The DOI matches its Crossref record; 50 of the 50 references it cites were found in a registry or an index; none of the 15 sources its claims cite is checked for support yet.
Quantum computations: algorithms and error correction
Open accessAbstract
Contents §0. Introduction §1. Abelian problem on the stabilizer §2. Classical models of computations 2.1. Boolean schemes and sequences of operations 2.2. Reversible computations §3. Quantum formalism 3.1. Basic notions and notation 3.2. Transformations of mixed states 3.3. Accuracy §4. Quantum models of computations 4.1. Definitions and basic properties 4.2. Construction of various operators from the elements of a basis 4.3. Generalized quantum control and universal schemes §5. Measurement operators §6. Polynomial quantum algorithm for the stabilizer problem §7. Computations with perturbations: the choice of a model §8. Quantum codes (definitions and general properties) 8.1. Basic notions and ideas 8.2. One-to-one codes 8.3. Many-to-one codes §9. Symplectic (additive) codes 9.1. Algebraic preparation 9.2. The basic construction 9.3. Error correction procedure 9.4. Torus codes §10. Error correction in the computation process: general principles 10.1. Definitions and results 10.2. Proofs §11. Error correction: concrete procedures 11.1. The symplecto-classical case 11.2. The case of a complete basis Bibliography
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
86 No basis given · 26 Own interpretation · 12 Not checked yet · 5 Own result · 2 Own method
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“We shall see that Q is a complete basis in the approximate sense, see Theorem 4.8 below.”
§ Quantum computations: algorithms and error correction > Choice of the basis.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The norm of an admissible transformation is always equal to one.”
§ Quantum computations: algorithms and error correction > 8 below).
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Proposition 2.3.We assume that F : Ν ->• B m (N C B n ) is computable in a basis A by means of a Boolean scheme of dimension L. Then F® can be represented in the basis A& = {/® : / G A} by a reversible scheme of dimension 2L + m.”
§ Quantum computations: algorithms and error correction > 2.2. Reversible computations.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Corollary. C^ is a complete basis for reversible computations.”
§ Quantum computations: algorithms and error correction > Corollary. C^ is a complete basis for reversible computations.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Proposition 3.6.Let Τ £ T(M,M ® 1) be an admissible transformation such that Try Γ = I M .Then there exists λ G D(J) such that TX = X <g> λ (for any X € L(M)).”
§ Quantum computations: algorithms and error correction > By Lemma 3.2 the non-zero eigenvalues of
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Any classical operator can be represented in the basis 3? = {1.3,σχ ,@,Λφ}.(This follows from the completeness of the classical reversible basis 31.)It follows that any classical operator can be represented in the basis Q.”
§ Quantum computations: algorithms and error correction > 4.2. Construction of various operators from the elements of a basis.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If a basis of the subgroup Ρ C Z 2 is known, then it is easy to find an element of the form (τη, -1) G P. Then ( m = g, that is, m is the discrete logarithm of g with base ζ.”
§ Quantum computations: algorithms and error correction > §1. Abelian problem on the stabilizer
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: The action of the group G on the finite set M and the group operations in G are easy to compute.
- The authors state this as an assumption: The group G in the stabilizer problem is Abelian.
- The authors state this as an assumption: The set M can be identified with a subset of the Boolean cube B^n.
- The authors state this as an assumption: The function F is available as an oracle or black box that returns F(g,x) for inputs in its domain.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The operators 5 and Κ (see (24)) generate a finite group Ε C U(25) of order 192.”
§ Quantum computations: algorithms and error correction > 4.2. Construction of various operators from the elements of a basis.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Moreover, there exists a rather simple polynomial algorithm for reducing a basis to a unique (for a given subgroup A C Z k ) canonical form.”
§ Quantum computations: algorithms and error correction > §1. Abelian problem on the stabilizer
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The authors state this as an assumption: The action of the group G on the finite set M and the group operations in G are easy to compute.
- The authors state this as an assumption: The group G in the stabilizer problem is Abelian.
- The authors state this as an assumption: The set M can be identified with a subset of the Boolean cube B^n.
- The authors state this as an assumption: The function F is available as an oracle or black box that returns F(g,x) for inputs in its domain.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Then the set of operators Η U V~1HV generates the whole group SU(M).”
§ Quantum computations: algorithms and error correction > (29) v2
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
- It may state two separate claims as one. 32
- Probably not a substantial claim. 19
- The quote only partly states this claim. 8
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Factorizing Ε relative to its centre {( k : k = 0,... , 7}, we obtain the corresponding group of unitary transformations ESp 2 (l) C UT(£) of order 24.8 The group ESp 2 (l) is isomorphic to the group of characteristic rotations of the cube, the Pauli matrices σ χ ,σ ν ,σ Ζ £ L(!B® n ) transforming like the basis unit vectors.”
§ Quantum computations: algorithms and error correction > 4.2. Construction of various operators from the elements of a basis.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Quantum probability is additive, just as classical probability is.”
§ Quantum computations: algorithms and error correction > 3.1. Basic notions and notation.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is clear that any Boolean scheme of dimension L can be transformed into a sequence of cL operations (where c is the largest number of bits acted upon by a basis function).Conversely, any sequence of L operations can be modelled by a Boolean scheme of dimension L.”
§ Quantum computations: algorithms and error correction > Λ[6;4,5] -[5] V[3,5;3] V[4;l,2] Λ [3; 3,4] V[2,4;2] V[l,3;l].
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is obvious that X),e Q P(p, Vj) + P(p, V±) = 1, where V± is the orthogonal complement of φ^€η V^•.”
§ Quantum computations: algorithms and error correction > 3.1. Basic notions and notation.
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“The discrete logarithm and decomposition into prime factors can be reduced to this problem.”
- Also stated in: § §1. Abelian problem on the stabilizer · plain
“The computational problems of 'factorization' and the 'discrete logarithm' can be reduced to an Abelian problem on the stabilizer.”
§ Quantum computations: algorithms and error correction > Abstract
What the sources say
- Not checked yetcites [49]
- Not checked yetcites [11]
plaindirectn = 2- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Also stated in: § §1. Abelian problem on the stabilizer · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“1) ΤΓ(ΓΧ) = Tr X for any X e L(W) or {which is equivalent) T*1 M = 1 N ;”
- Also stated in: § 3.2. Transformations of mixed states. · plain
“3) Τ is a positive operator.”
§ Quantum computations: algorithms and error correction > 3.2. Transformations of mixed states.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
- Also stated in: § 3.2. Transformations of mixed states. · plain
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“First, any mixed state ρ £ D(N) can be represented as”
§ Quantum computations: algorithms and error correction > 3.1. Basic notions and notation.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It follows that St_F(a) is isomorphic to Z k and has a basis {51,... ,#&} of polynomial dimension, that is, ]Cj = i size(gj-) ^ poly(n + k).”
§ Quantum computations: algorithms and error correction > §1. Abelian problem on the stabilizer
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: The action of the group G on the finite set M and the group operations in G are easy to compute.
- The authors state this as an assumption: The group G in the stabilizer problem is Abelian.
- The authors state this as an assumption: The set M can be identified with a subset of the Boolean cube B^n.
- The authors state this as an assumption: The function F is available as an oracle or black box that returns F(g,x) for inputs in its domain.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The basis Q' generates a set of operators closed under the transformation U >-¥ A(U) (and also under U i-> U~1).”
§ Quantum computations: algorithms and error correction > (29) v2
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Unfortunately, Λ(5) can only be represented approximately, which gives rise to some technical inconvenience.”
§ Quantum computations: algorithms and error correction > {|°^ * a = ° (28)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“These operators commute with one another and have eigenvalues |(1 ± %/-15), which are not roots of 1.”
§ Quantum computations: algorithms and error correction > (29) v2
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The canonical generators are unitary, they belong to the gradation spaces, and have the following properties:”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The information operators are defined to within multiplication by a verification operator.”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Besides, each element of Q' can be represented in the basis Q.”
§ Quantum computations: algorithms and error correction > (29) v2
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A symplectic code can be completely characterized by specifying the verification operators Χχ,... ,X S and information operators Υ Ί & = (U • £/t)a T [s + k]”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Corollary.The operator Λ(Τί) {see Example 4.10) can be represented with accuracyfoot_10 Ο(δ) by a scheme of dimension poly(log(l/(5)) in the basis Q.This scheme can be constructed algorithmically in time poly(log(l/(5)).”
§ Quantum computations: algorithms and error correction > U L :
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“(Warning: the operator @ copies each basis state appearing in the quantum superposition, rather than the superposition as a whole!)”
§ Quantum computations: algorithms and error correction > Modelling of classical probability. (Compare with Example 3.3.)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is not hard to show that ESp 2 (n) is generated by the unitary operators S, Κ and ©, which can be applied to different qbits.”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The operators σ {αχ βχ,... ,α η β η ) = σ αι β 1 (&-• -®σ αη β η form the standard basis in L(S !8ln ).Each of them is both unitary and Hermitian.Their eigenvalues are equal to ±1.”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It follows that GL 2 (n) g ESp 2 (n).The group GL 2 (n) is generated by the operator ©, which can be applied to various qbits.”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is obvious that w(/i,/j) = 0 (i,j = 1,2,3,4).linearly independent, so E(A, 2) U F + = {0}.”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If t is a unitary transformation, then the condition of (2, 2Γ, Testability can be replaced by (2, Γ, Testability.For classical schemes it suffices to require (Ι,Γ, Testability.”
§ Quantum computations: algorithms and error correction > Remark.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Theorem 4 . 8 . Any unitary matrix U £ U(!B® n ) can be represented with arbitrary accuracy δ > 0 by a quantum scheme of dimension exp(O(n)) poly(log(l/i)) in the basis Q'.This scheme can be constructed algorithmically in time exp(O(n))poly(log(l/i)).”
§ Quantum computations: algorithms and error correction > QB[0] (S-S)[0] R<-[0;A O } T'[A 0 , A U A 2 ,A 3 ] {\ξ)(ξ\)[Αο\.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Error correction, that is, the computation of v, can be reduced to finding the smallest 1-chain given its boundary.In turn, this problem can be reduced to the problem of least weight matching, for which a polynomial algorithm is known [55].”
§ Quantum computations: algorithms and error correction > 9.4.
What the sources say
- Not checked yetcites [55]
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It follows that 2s old errors (which existed prior to applying the procedure Σ) in the code qbits will be corrected and only 0(1) ^ β' new ones will appear.”
§ Quantum computations: algorithms and error correction > 11.1. The symplecto-classical case.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: In the symplecto-classical case, all classical elements in the basis SC are accurate.
- The authors state this as an assumption: The classical scheme Σ10 is absolutely reliable.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Suppose that two errors occur as the scheme Σ is run.We can always find an i such that the subschemes Σί,Σ( + ι,Σ; + 2 work correctly.It follows that hi = hi + i = hi+2-”
§ Quantum computations: algorithms and error correction > 11.1. The symplecto-classical case.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: In the symplecto-classical case, all classical elements in the basis SC are accurate.
- The authors state this as an assumption: The classical scheme Σ10 is absolutely reliable.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“To represent the transformation Λ in the code REP(n, 1) (η ^ 5) it suffices to perform the operation Λ for each pair of code bits, after correcting any possible errors in the input (see Fig. 2b).If a new error occurs (in one of the subschemes R. n or one of the elements Λ), then no more than one output bit will be corrupted.”
§ Quantum computations: algorithms and error correction > Additional information
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It follows that if the state |i7 + ) is prepared in the second qbit, then W acts on the first qbit in the same way as Κ.”
§ Quantum computations: algorithms and error correction > Stable realization of the operator K.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is easily seen that U • U^ is a symplectic transformation.”
§ Quantum computations: algorithms and error correction > Ρ(\ξ,ξ,Ο(ξ,ξ,ξ\) = \{\Φ + )(Φ+\,ο) + i(|V->W-l,i),
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This situation can be rectified using the identity A C (A C (U))[1,2,A}=A C (U)[3,A}A{3;1,2).”
§ Quantum computations: algorithms and error correction > Stable realization of the operator K.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: unclear — It is unclear why the paper says it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Let D o be the identity code and let D r+l = C <v D r (r = 0,1,...).(The codes D r are called cascade codes.)In the basis A we shall construct schemes Φ ίΓ (t £ .A) with the following property: any transformation Τ e Trans(* ir ,(5) represents t in the code D r with accuracy e r = co^l 2 ^, where c = const.Moreover, we require that any Τ £ Trans($/ r ,<5) should represent / in D in the strong”
§ Quantum computations: algorithms and error correction > Remark.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The identity transformation i^ will be realized by a (2, 2F,F')-stable scheme Σ.All the remaining elements of the basis will be realized in a (2, Γ', Testable way.”
§ Quantum computations: algorithms and error correction > 11.1. The symplecto-classical case.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: In the symplecto-classical case, all classical elements in the basis SC are accurate.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It is easy to verify that the transformations Υ α β Ί • Υ&β Ί are symplectic, so they can be realized in a stable way.”
§ Quantum computations: algorithms and error correction > Additional information
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Nevertheless (this is a key point in the whole construction!), the depth of the scheme Σ9 ο • • • ο Σι is bounded because TOR(fc) are codes with local verification.Consequently, each of the two possible errors can corrupt only 0(1) qbits.”
§ Quantum computations: algorithms and error correction > 11.1. The symplecto-classical case.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: In the symplecto-classical case, all classical elements in the basis SC are accurate.
- The authors state this as an assumption: The classical scheme Σ10 is absolutely reliable.
In the paper
“He proved that a computation of length L can be carried out if the accuracy of each operation is (logL)~c (c = const).”
§ Quantum computations: algorithms and error correction > §0. Introduction
What the sources say
- Not checked yetcites [43]
plainconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“a quantum state cannot be copied”
§ Quantum computations: algorithms and error correction > §0. Introduction
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Since any finitely-generated Abelian group is a factor group Z fc , we can assume without loss of generality that G = Z*.”
§ Quantum computations: algorithms and error correction > §1. Abelian problem on the stabilizer
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
- The authors state this as an assumption: The group G in the stabilizer problem is Abelian.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Therefore, there is a one-to-one correspondence between combined spaces and finite-dimensional C*-algebras.”
§ Quantum computations: algorithms and error correction > 3.1. Basic notions and notation.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The less general case G = (Z p )* was studied by Grigoriev in [14], where he constructed an algorithm that is polynomial in k and ρ (but not in logp).”
§ Quantum computations: algorithms and error correction > §1. Abelian problem on the stabilizer
What the sources say
- Not checked yetcites [14]
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: The group G in the stabilizer problem is Abelian.
In the paper
“In 1994 Shor designed polynomial quantum algorithms for the discrete logarithm and the decomposition of integers into prime factors [11].”
§ Quantum computations: algorithms and error correction > §0. Introduction
What the sources say
- Not checked yetcites [11]
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Both problems are regarded as difficult.”
§ Quantum computations: algorithms and error correction > §0. Introduction
What the sources say
- Not checked yetcites [12]
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the literal sense this is impossible.”
§ Quantum computations: algorithms and error correction > §7. Computations with perturbations: the choice of a model
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“Symplectic codes were first defined in [29].Quasilinear codes were considered (in slightly different terms) by Steane [20], and Shor and Calderbank [19].”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
What the sources say
- Not checked yetcites [29]
- Not checked yetcites Steane [20]
- Not checked yetcites Shor and Calderbank [19]
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is clear that the computation of Sti'(o) is polynomially equivalent to that of H.”
§ Quantum computations: algorithms and error correction > §6. Polynomial quantum algorithm for the stabilizer problem
boostedcorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Theorem 8.3.Let 5H C Τ(Ή, Ή') be a set of completely positive transformations.We define the minimal space of operators £ C L(N, N') such that 91 C £ • £*.The following conditions are equivalent:”
§ Quantum computations: algorithms and error correction > 8.2. One-to-one codes.
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
The quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The group Η is generated by sufficiently many random elements (see below).”
§ Quantum computations: algorithms and error correction > §6. Polynomial quantum algorithm for the stabilizer problem
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the problem concerning the transmission of information (or a quantum state) many-to-one codes provide no advantage as compared to one-to-one codes.”
§ Quantum computations: algorithms and error correction > 8.2. One-to-one codes.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“The construction described is due to Shor [43].”
§ Quantum computations: algorithms and error correction > 11.2. The case of a complete basis.
What the sources say
- Not checked yetcites Shor [43]
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The only way around this difficulty is to supply the input information in an encoded form.The output information should also be encoded using a many-to-one code.(The ambiguity includes errors arising at the last step.)Finally, it is natural to encode bitwise all the intermediate results obtained as the accurate scheme is run.”
§ Quantum computations: algorithms and error correction > 10.1. Definitions and results.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“However, if R is realized in an unstable way, the error involved in it can only corrupt the classical bit 2, but not the quantum bit 1 (represented in the code C).Thus we must repeat the measurement five times and take the MAJ function of the five results.”
§ Quantum computations: algorithms and error correction > Stable realization of the operator K.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
In the paper
“The scheme (64) is almost stable: two errors can corrupt only the classical bit 0. Therefore, the measurement is to be repeated five times, followed by taking the MAJ function.”
§ Quantum computations: algorithms and error correction > QB[0] (S-S)[0] R<-[0;A O } T'[A 0 , A U A 2 ,A 3 ] {\ξ)(ξ\)[Αο\.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“(This is always enough, since we assume that no more than two errors can occur.)”
§ Quantum computations: algorithms and error correction > Stable realization of the state |0).
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
- The authors state this as an assumption: No more than two errors can occur.
The authors state this as an assumption.
The action of the group G on the finite set M and the group operations in G are easy to compute.
“We assume that this action and the group operations in G are easy to compute.”
The authors state this as an assumption.
The group G in the stabilizer problem is Abelian.
“We shall assume instead that G is an Abelian group.”
The authors state this as an assumption.
The set M can be identified with a subset of the Boolean cube B^n.
“we shall assume that Μ is identified with a subset of the Boolean cube”
The authors state this as an assumption.
The function F is available as an oracle or black box that returns F(g,x) for inputs in its domain.
“The function F is to be regarded as an oracle, that is, a 'black box' which receives an input word”
The paper relies on this without support.
In real-world conditions, qubits are subject to perturbations even when no operation is performed on them.
“Under real-life conditions qbits will be subject to perturbations even if nothing is done to them.”
The paper relies on this without support.
In sensible applications, the function F can be computed in polynomial time.
“In all sensible applications (see the examples below) F can be computed in polynomial time.”
The paper relies on this without support.
Operators acting on a small number of quantum bits can be physically realized.
“It is possible to realize such operators physically if η is small.”
The authors state this as an assumption.
Errors at different steps are statistically independent because the environment is renewed each step.
“Assuming that the environment is renewed at each step, we can exclude it from consideration and use the language of transformations of density operators (see §3.2). (This assumption means that errors at different steps are statistically independent.)”
The authors state this as an assumption.
Perturbations leave classical bits in classical states.
“Nevertheless, it is natural to assume that the perturbation leaves the classical bits in a classical state.”
The authors state this as an assumption.
The accuracy constant δ is small enough and does not depend on the specific layered scheme.
“The accuracy constant δ will be assumed to be sufficiently small, but independent of the concrete layered scheme.”
The authors state this as an assumption.
The quantum computer operates with mixed states, and unperturbed elementary transformations need not be of the form U·U†.
“It proves convenient to assume from the very beginning that the quantum computer works with mixed states, and even that unperturbed elementary transformations do not necessarily have the form U • t/t.Such computations are called mixed.”
The authors state this as an assumption.
In the simplified quantum code problem, a perturbation occurs at only one step.
“Suppose that a perturbation acts at a single step only.Arbitrary transformations, namely, coding and decoding, are allowed before and after this step.”
The authors state this as an assumption.
The quantum channel belongs to a class where the transformation is a tensor product of single-qubit admissible transformations, each within δ of the identity.
“We shall consider the following class of channels: Ν = DM' = T> n , Τ = Τι®• • •®Γ η , where Tj are admissible transformations with a single qbit”
The authors state this as an assumption.
There exists a T in the error set such that ||T - f|| < ε.
“We assume that there is a T £ 9t such that”
The authors state this as an assumption.
In the derived code setting, Ή'=3Γ and £ contains 3j.
“As a rule, we shall assume that”
The authors state this as an assumption.
In the symplecto-classical case, all classical elements in the basis SC are accurate.
“Now we shall assume that all the classical elements in the basis SC are accurate.”
The authors state this as an assumption.
The classical scheme Σ10 is absolutely reliable.
“the classical scheme Σ10 is assumed to be absolutely reliable.”
The authors state this as an assumption.
Until concrete examples are reached, all bits are assumed to be quantum bits.
“Until we arrive at concrete examples we shall assume that all bits are quantum ones.”
The authors state this as an assumption.
No more than two errors can occur.
“(This is always enough, since we assume that no more than two errors can occur.)”
The authors state this as an assumption.
The first step, construction of schemes similar to R_n, has already been made.
“(It is assumed that the first step, namely, the construction of schemes similar to R n , has already been made.)”
The authors state this as an assumption.
All spaces considered are unitary.
“For simplicity we shall assume that all the spaces considered are unitary.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“The idea is based on a theorem by Schumacher and Nielsen[61] which asserts that a quantum state can be corrected only if no information about the state has leaked to the environment through the noise process.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [61]
In the citing paper
“More precisely, if the reduced density matrix on any t qubits does not depend on the logical qubit, then there exists a unitary operation which recovers the original state even if the environment interacted with t qubits, i.e. t errors occurred.”
From Fault-Tolerant Quantum Computation with Constant Error Rate (2008) · cites this paper as [61]
In the citing paper
“If the EPG is smaller than a threshold, then scalable quantum computing is possible. 5-8”
From Quantum computing with realistically noisy devices (2005) · cites this paper as [6]
In the citing paper
“Estimated thresholds vary from below 10 -6 [5-8] to 3 × 10 -3 [9], with 10 -4 [10] often quoted as the EPG to be achieved in experimental quantum computing.”
From Quantum computing with realistically noisy devices (2005) · cites this paper as [6]
Citations
Cites
New directions in cryptography
Diffie, W., & Hellman, M. (1976) · IEEE Transactions on Information Theory · doi:10.1109/tit.1976.1055638 · Cited by 14,713
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels
Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., & Wootters, W. K. (1993) · Physical Review Letters · doi:10.1103/physrevlett.70.1895 · Cited by 13,703
The Theory of Error-Correcting Codes
(1977) · North-Holland mathematical library · doi:10.1016/s0924-6509(08)x7030-8 · Cited by 10,891
Combinatorial Optimization: Algorithms and Complexity.
Johnson, D. S., Papadimitriou, C. H., & Steiglitz, K. (1984) · American Mathematical Monthly · doi:10.2307/2322374 · Cited by 6,007
Mixed-state entanglement and quantum error correction
Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review A · doi:10.1103/physreva.54.3824 · Cited by 5,272
Quantum theory, the Church–Turing principle and the universal quantum computer
Deutsch, D. (1985) · Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences · doi:10.1098/rspa.1985.0070 · Cited by 4,597
Scheme for reducing decoherence in quantum computer memory
Shor, P. W. (1995) · Physical Review A · doi:10.1103/physreva.52.r2493 · Cited by 4,479
Elementary gates for quantum computation
Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator, T., Smolin, J. A., & Weinfurter, H. (1995) · Physical Review A · doi:10.1103/physreva.52.3457 · Cited by 4,311
Logical Reversibility of Computation
Bennett, C. H. (1973) · IBM Journal of Research and Development · doi:10.1147/rd.176.0525 · Cited by 3,679
Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
Bennett, C. H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J. A., & Wootters, W. K. (1996) · Physical Review Letters · doi:10.1103/physrevlett.76.722 · Cited by 2,961
Research works printed but not linked: 6
- D. Grigoriev, "Testing shift equivalence of polynomials using quantum mechanics", Preprint, 1995.
- Β. Β. Plenio, V. Vedral, and P. L. Knight, "Optimal realistic quantum error correcting code", LANL e-print quant-ph/9603022, http://xxx.lanl.gov. · arXiv
- C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, "Mixed state entanglement and quantum error-correcting codes", LANL e-print quant-ph/9604024, http://xxx.lanl.gov. · arXiv
- A. S. Kholevo, "Some estimates for the amount of information transmittable by a quantum communication channel", Problemy Peredachi Informatsii 9:3 (1973), 3-11; English transl. in Problems of Information Transmission 9 (1973).
- A. Yu. Kitaev, "Fault-tolerant quantum computation by anions", LANL e-print quant-ph/9707021, http://xxx.lanl.gov. · arXiv
- Y. Lecerf, "Machines de Turing reversibles. Recursive insolubilite en η 6 Ν de l'equation u = Θ" ou θ est un 'isomorphisme de codes'", C.R. Acad. Sci. Paris 257 (1963), 2597-2600.
Cited by
Sorted by citations
Items 201–300 of 1,002
Topological Quantum Computation
Ogburn, R. W., & Preskill, J. P. (1999) · Lecture notes in computer science · doi:10.1007/3-540-49208-9_31 · Cited by 75
Quantum Error Correction Decoheres Noise
Beale, S. J., Wallman, J. J., Gutiérrez, M., Brown, K. R., & Laflamme, R. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.190501 · Cited by 84
Resource theory of asymmetric distinguishability for quantum channels
Wang, X., & Wilde, M. M. (2019) · Physical Review Research · doi:10.1103/physrevresearch.1.033169 · Cited by 96
Quantum process tomography with coherent states
Rahimi-Keshari, S., Scherer, A., Mann, A., Rezakhani, A. T., Lvovsky, A. I., & Sanders, B. C. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/1/013006 · Cited by 76
Entanglement Cost of Quantum Channels
Berta, M., Brandão, F. G. S. L., Christandl, M., & Wehner, S. (2013) · IEEE Transactions on Information Theory · doi:10.1109/tit.2013.2268533 · Cited by 80
Quantum Algorithm for Calculating Molecular Vibronic Spectra
Sawaya, N. P. D., & Huh, J. (2019) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.9b01117 · Cited by 79
Chamberland, C., & Noh, K. (2020) · npj Quantum Information · doi:10.1038/s41534-020-00319-5 · Cited by 97
Quantum Computation with Topological Codes
Fujii, K. (2015) · SpringerBriefs in mathematical physics · doi:10.1007/978-981-287-996-7 · Cited by 79
Statistical speed of quantum states: Generalized quantum Fisher information and Schatten speed
Gessner, M., & Smerzi, A. (2018) · Physical Review A · doi:10.1103/physreva.97.022109 · Cited by 73
Information-theoretic aspects of the generalized amplitude-damping channel
Khatri, S., Sharma, K., & Wilde, M. M. (2020) · Physical Review A · doi:10.1103/physreva.102.012401 · Cited by 86
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
Wocjan, P., & Yard, J. (2008) · Quantum Information and Computation · doi:10.26421/qic8.1-2-10 · Cited by 76
Resource-Optimal Single-Qubit Quantum Circuits
Bocharov, A., & Svore, K. M. (2012) · Physical Review Letters · doi:10.1103/physrevlett.109.190501 · Cited by 69
Liquid-state NMR simulations of quantum many-body problems
Negrevergne, C., Somma, R. D., Ortíz, G., Knill, E., & Laflamme, R. (2005) · Physical Review A · doi:10.1103/physreva.71.032344 · Cited by 71
Quantum Computation and the Localization of Modular Functors
Freedman, M. H. (2001) · Foundations of Computational Mathematics · doi:10.1007/s102080010006 · Cited by 72
Protecting geometric gates by dynamical decoupling
Xu, G., & Long, G. (2014) · Physical Review A · doi:10.1103/physreva.90.022323 · Cited by 68
Optimization of a solid-state electron spin qubit using gate set tomography
Dehollain, J. P., Muhonen, J. T., Blume-Kohout, R., Rudinger, K. M., Gamble, J. K., Nielsen, E., Laucht, A., Simmons, S., Kalra, R., Dzurak, A. S., et al. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/10/103018 · Cited by 72
Geometric Rényi Divergence and its Applications in Quantum Channel Capacities
Fang, K., & Fawzi, H. (2021) · Communications in Mathematical Physics · doi:10.1007/s00220-021-04064-4 · Cited by 87
Quantum Algorithm for Petz Recovery Channels and Pretty Good Measurements
Gilyén, A., Lloyd, S., Marvian, I., Quek, Y., & Wilde, M. M. (2022) · Physical Review Letters · doi:10.1103/physrevlett.128.220502 · Cited by 67
Resource theory of asymmetric distinguishability for quantum channels
Wang, X., & Wilde, M. M. (2019) · Civil War Book Review · Cited by 67
Characterizing Quantum Dynamics with Initial System-Environment Correlations
Ringbauer, M., Wood, C. J., Modi, K. K., Gilchrist, A., White, A. G., & Fedrizzi, A. (2015) · Physical Review Letters · doi:10.1103/physrevlett.114.090402 · Cited by 67
Quantum Algorithms for Hidden Nonlinear Structures
Childs, A. M., Schulman, L. J., & Vazirani, U. V. (2007) · Annual Symposium on Foundations of Computer Science · doi:10.1109/focs.2007.18 · Cited by 67
Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits
Xu, Q., Zheng, G., Wang, Y., Zoller, P., Clerk, A. A., & Jiang, L. (2023) · npj Quantum Information · doi:10.1038/s41534-023-00746-0 · Cited by 77
Bulk-edge correspondence of one-dimensional quantum walks
Cedzich, C., Grünbaum, F. A., Stahl, C., Velázquez, L., Werner, A. H., & Werner, R. F. (2016) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/49/21/21lt01 · Cited by 70
Duality of Quantum and Classical Error Correction Codes: Design Principles and Examples
Babar, Z., Chandra, D., Nguyen, H. V., Botsinis, P., Alanis, D., Ng, S. X., & Hanzo, L. (2019) · IEEE Communications Surveys & Tutorials · doi:10.1109/comst.2018.2861361 · Cited by 65
Quantum singular value decomposer
Bravo-Prieto, C., García-Martín, D., & Latorre, J. I. (2020) · Physical Review A · doi:10.1103/physreva.101.062310 · Cited by 64
Extending quantum probabilistic error cancellation by noise scaling
Mari, A., Shammah, N., & Zeng, W. J. (2021) · Physical Review A · doi:10.1103/physreva.104.052607 · Cited by 76
Application of quantum approximate optimization algorithm to job shop scheduling problem
Kurowski, K., Pecyna, T., Slysz, M., Różycki, R., Waligóra, G., & Węglarz, J. (2023) · European Journal of Operational Research · doi:10.1016/j.ejor.2023.03.013 · Cited by 72
Decoupling with unitary approximate two-designs
Szehr, O., Dupuis, F., Tomamichel, M., & Renner, R. (2013) · New Journal of Physics · doi:10.1088/1367-2630/15/5/053022 · Cited by 71
Fault-tolerant quantum computation versus Gaussian noise
Ng, H. K., & Preskill, J. P. (2009) · Physical Review A · doi:10.1103/physreva.79.032318 · Cited by 64
Nishimura, H., & Ozawa, M. (2002) · Theoretical Computer Science · doi:10.1016/s0304-3975(01)00111-6 · Cited by 71
Quantum Computation Beyond the Circuit ModelPreprint
Jordan, S. P. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0809.2307 · Cited by 69
Decoding algorithms for surface codes
deMarti iOlius, A., Fuentes, P., Orús, R., Crespo, P. M., & Etxezarreta Martinez, J. (2024) · Quantum · doi:10.22331/q-2024-10-10-1498 · Cited by 60
Quantum error correction with the semion code
Dauphinais, G., Ortíz, L., Varona, S., & Martin-Delgado, M. A. (2019) · New Journal of Physics · doi:10.1088/1367-2630/ab1ed8 · Cited by 65
Sufficient condition on noise correlations for scalable quantum computing
Preskill, J. P. (2013) · Quantum Information and Computation · doi:10.26421/qic13.3-4-1 · Cited by 77
Wavefunctions for topological quantum registers
Ardonne, E., & Schoutens, K. (2006) · Annals of Physics · doi:10.1016/j.aop.2006.07.015 · Cited by 63
Sharma, K., Wilde, M. M., Adhikari, S., & Takeoka, M. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aac11a · Cited by 66
Quantum inverse iteration algorithm for programmable quantum simulators
Kyriienko, O. (2020) · npj Quantum Information · doi:10.1038/s41534-019-0239-7 · Cited by 66
Williams, C. P. (2010) · Texts in computer science · doi:10.1007/978-1-84628-887-6_2 · Cited by 68
Asthana, A., Kumar, A., Abraham, V., Grimsley, H. R., Zhang, Y., Cincio, Ł., Tretiak, S., Dub, P. A., Economou, S. E., Barnes, E., et al. (2023) · Chemical Science · doi:10.1039/d2sc05371c · Cited by 65
Yoshida, B. (2010) · Annals of Physics · doi:10.1016/j.aop.2010.10.009 · Cited by 61
Modeling quantum noise for efficient testing of fault-tolerant circuits
Magesan, E., Puzzuoli, D., Granade, C., & Cory, D. G. (2013) · Physical Review A · doi:10.1103/physreva.87.012324 · Cited by 69
Optimized surface code communication in superconducting quantum computers
Javadi-Abhari, A., Gokhale, P., Holmes, A. A., Franklin, D. M., Brown, K. R., Martonosi, M., & Chong, F. T. (2017) · IEEE/ACM International Symposium on Microarchitecture (MICRO) · doi:10.1145/3123939.3123949 · Cited by 69
Lidar, D. A., Bacon, D., Kempe, J., & Whaley, K. B. (2001) · Physical Review A · doi:10.1103/physreva.63.022307 · Cited by 62
Identifying the Information Gain of a Quantum Measurement
Berta, M., Renes, J. M., & Wilde, M. M. (2014) · IEEE Transactions on Information Theory · doi:10.1109/tit.2014.2365207 · Cited by 60
Quantum Coding with Low-Depth Random Circuits
Gullans, M. J., Krastanov, S., Huse, D. A., Jiang, L., & Flammia, S. T. (2021) · Physical Review X · doi:10.1103/physrevx.11.031066 · Cited by 68
Arbitrary two-qubit computation in 23 elementary gates
Bullock, S. S., & Markov, I. L. (2003) · Physical Review A · doi:10.1103/physreva.68.012318 · Cited by 59
Scalability of Shor’s algorithm with a limited set of rotation gates
Fowler, A. G., & Hollenberg, L. C. L. (2004) · Physical Review A · doi:10.1103/physreva.70.032329 · Cited by 59
Algorithmic Complexity and Entanglement of Quantum States
Mora, C. E., & Briegel, H. J. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.200503 · Cited by 59
Anticoncentration theorems for schemes showing a quantum speedup
Hangleiter, D., Bermejo-Vega, J., Schwarz, M., & Eisert, J. (2018) · Quantum · doi:10.22331/q-2018-05-22-65 · Cited by 58
Ding, Y., & Chong, F. T. (2020) · Synthesis lectures on computer architecture · doi:10.2200/s01014ed1v01y202005cac051 · Cited by 63
Quantum Error Correction and Fault Tolerant Quantum Computing
Grassl, M., & Rötteler, M. (2009) · Springer eBooks · doi:10.1007/978-0-387-30440-3_435 · Cited by 58
Approaches to Quantum Error Correction
Kempe, J. (2006) · Birkhäuser Basel eBooks · doi:10.1007/978-3-7643-7808-0_3 · Cited by 57
Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations
Zhang, J., McClean, J. R., Babbush, R., & Neven, H. (2019) · Physical Review Applied · doi:10.1103/physrevapplied.12.064041 · Cited by 64
Quantum Channel Simulation and the Channel’s Smooth Max-Information
Fang, K., Wang, X., Tomamichel, M., & Berta, M. (2019) · IEEE Transactions on Information Theory · doi:10.1109/tit.2019.2943858 · Cited by 57
Approximate Counting and Quantum Computation
Bordewich, M., Freedman, M. H., Lovász, L., & Welsh, D. J. A. (2005) · Combinatorics Probability Computing · doi:10.1017/s0963548305007005 · Cited by 62
Stabilizer entropy dynamics after a quantum quench
Rattacaso, D., Leone, L., Oliviero, S. F. E., & Hamma, A. (2023) · Physical Review A · doi:10.1103/physreva.108.042407 · Cited by 67
Fidelity and Fisher information on quantum channels
Yuan, H., & Fung, C. F. (2017) · New Journal of Physics · doi:10.1088/1367-2630/aa874c · Cited by 55
A Hybrid Quantum-Classical Approach to Mitigating Measurement Errors in Quantum Algorithms
Kwon, H., & Bae, J. (2021) · IEEE Transactions on Computers · doi:10.1109/tc.2020.3009664 · Cited by 54
Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral
Gustafson, E., Lamm, H., Lovelace, F., & Musk, D. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.106.114501 · Cited by 71
Quantum computing with and for many-body physics
Ayral, T., Besserve, P., Denis Lacroix, A., & Guzman, E. A. R. (2023) · The European Physical Journal A · doi:10.1140/epja/s10050-023-01141-1 · Cited by 71
Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation
Yamasaki, H., & Koashi, M. (2024) · Nature Physics · doi:10.1038/s41567-023-02325-8 · Cited by 73
Simulation of rare events in quantum error correction
Bravyi, S. B., & Vargo, A. (2013) · Physical Review A · doi:10.1103/physreva.88.062308 · Cited by 68
Algorithms for the Optimization of Quantum Circuits
Amy, M. (2013) · UWSpace (University of Waterloo) · Cited by 54
An Improved Implementation Approach for Quantum Phase Estimation on Quantum Computers
Mohammadbagherpoor, H., Oh, Y.-H., Dreher, P., Singh, A., Yu, X., & Rindos, A. (2019) · IEEE International Conference on Rebooting Computing (ICRC) · doi:10.1109/icrc.2019.8914702 · Cited by 53
Advancing hybrid quantum–classical computation with real-time execution
Lubinski, T., Granade, C., Anderson, A., Geller, A., Roetteler, M., Petrenko, A., & Heim, B. (2022) · Frontiers in Physics · doi:10.3389/fphy.2022.940293 · Cited by 63
Shorter gate sequences for quantum computing by mixing unitaries
Campbell, E. T. (2017) · Physical Review A · doi:10.1103/physreva.95.042306 · Cited by 74
Kleißler, F., Lazariev, A., & Arroyo-Camejo, S. (2018) · npj Quantum Information · doi:10.1038/s41534-018-0098-7 · Cited by 56
Theory of robust multiqubit nonadiabatic gates for trapped ions
Shapira, Y., Shaniv, R., Manovitz, T., Akerman, N., Peleg, L., Gazit, L., Ozeri, R., & Stern, A. (2020) · Physical Review A · doi:10.1103/physreva.101.032330 · Cited by 54
Identification and Distance Measures of Measurement Apparatus
Ji, Z., Feng, Y., Duan, R., & Ying, M. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.200401 · Cited by 52
Quantum Computational Complexity
Watrous, J. (2011) · Springer eBooks · doi:10.1007/978-1-4614-1800-9_147 · Cited by 55
Tyson, J. (2010) · Journal of Mathematical Physics · doi:10.1063/1.3463451 · Cited by 57
Semidefinite programs for completely bounded normsPreprint
Watrous, J. (2009) · arXiv (Cornell University) · doi:10.48550/arxiv.0901.4709 · Cited by 52
Quantum Simulation of Dissipative Collective Effects on Noisy Quantum Computers
Cattaneo, M., Rossi, M. A. C., García-Pérez, G., Zambrini, R., & Maniscalco, S. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.010324 · Cited by 54
Information geometry of Gaussian channels
Monràs, A., & Illuminati, F. (2010) · Physical Review A · doi:10.1103/physreva.81.062326 · Cited by 52
Barz, S. (2015) · Journal of Physics B Atomic Molecular and Optical Physics · doi:10.1088/0953-4075/48/8/083001 · Cited by 56
Experimental magic state distillation for fault-tolerant quantum computing
Souza, A. M., Zhang, J., Ryan, C. A., & Laflamme, R. (2011) · Nature Communications · doi:10.1038/ncomms1166 · Cited by 51
Krotov method for optimal control of closed quantum systems
Morzhin, O. V., & Pechen, A. N. (2019) · Russian Mathematical Surveys · doi:10.1070/rm9835 · Cited by 59
Determining the proton content with a quantum computer
Pérez-Salinas, A., Cruz-Martinez, J. M., Alhajri, A., & Carrazza, S. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.103.034027 · Cited by 51
Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code
Paetznick, A., & Reichardt, B. W. (2012) · Quantum Information and Computation · doi:10.26421/qic12.11-12-10 · Cited by 70
Gate-error analysis in simulations of quantum computers with transmon qubits
Willsch, D., Nocon, M., Jin, F., Raedt, H. A. D., & Michielsen, K. F. L. (2017) · Physical Review A · doi:10.1103/physreva.96.062302 · Cited by 53
Ultimate Limits for Multiple Quantum Channel Discrimination
Zhuang, Q., & Pirandola, S. (2020) · Physical Review Letters · doi:10.1103/physrevlett.125.080505 · Cited by 50
Experimental Characterization of Crosstalk Errors with Simultaneous Gate Set Tomography
Rudinger, K., Hogle, C., Naik, R., Hashim, A., Lobser, D., Santiago, D., Grace, M., Nielsen, E., Proctor, T. J., Seritan, S., et al. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040338 · Cited by 50
On a measure of distance for quantum strategiesPreprint
Gutoski, G. (2016) · Cited by 50
Hunting for quantum-classical crossover in condensed matter problems
Yoshioka, N., Okubo, T., Suzuki, Y., Koizumi, Y., & Mizukami, W. (2024) · npj Quantum Information · doi:10.1038/s41534-024-00839-4 · Cited by 75
Overhead analysis of universal concatenated quantum codes
Chamberland, C., Jochym-O’Connor, T., & Laflamme, R. (2017) · Physical Review A · doi:10.1103/physreva.95.022313 · Cited by 52
Polynomial time quantum computation with advice
Nishimura, H., & Yamakami, T. (2004) · Information Processing Letters · doi:10.1016/j.ipl.2004.02.005 · Cited by 49
Performance of Planar Floquet Codes with Majorana-Based Qubits
Paetznick, A., Knapp, C., Delfosse, N., Bauer, B., Haah, J., Hastings, M. B., & Silva, M. P. D. (2023) · PRX Quantum · doi:10.1103/prxquantum.4.010310 · Cited by 48
Astrea: Accurate Quantum Error-Decoding via Practical Minimum-Weight Perfect-Matching
Vittal, S., Das, P., & Qureshi, M. K. (2023) · Proceedings - International Symposium on Computer Architecture/Proceedings · doi:10.1145/3579371.3589037 · Cited by 48
Entanglement cost and quantum channel simulation
Wilde, M. M. (2018) · Physical Review A · doi:10.1103/physreva.98.042338 · Cited by 53
Gaitan, F. (2021) · Advanced Quantum Technologies · doi:10.1002/qute.202100055 · Cited by 47
Operational distance and fidelity for quantum channels
Belavkin, V. P., D’Ariano, G. M., & Raginsky, M. (2005) · Journal of Mathematical Physics · doi:10.1063/1.1904510 · Cited by 47
Quantum computing and hidden variables
Aaronson, S. (2005) · Physical Review A · doi:10.1103/physreva.71.032325 · Cited by 47
Fundamental limits on quantum dynamics based on entropy change
Das, S., Khatri, S., Siopsis, G., & Wilde, M. M. (2018) · Journal of Mathematical Physics · doi:10.1063/1.4997044 · Cited by 52
Quantum-Fourier-transform-based quantum arithmetic with qudits
Pavlidis, A., & Floratos, E. G. (2021) · Physical Review A · doi:10.1103/physreva.103.032417 · Cited by 47
Quantum algorithms for approximate function loading
Marin-Sanchez, G., Gonzalez-Conde, J., & Sanz, M. (2023) · Physical Review Research · doi:10.1103/physrevresearch.5.033114 · Cited by 55
Continuous quantum error correction for non-Markovian decoherence
Oreshkov, O., & Brun, T. A. (2007) · Physical Review A · doi:10.1103/physreva.76.022318 · Cited by 46
Zero-Knowledge Proof Systems for QMA
Broadbent, A., Ji, Z., Song, F., & Watrous, J. (2016) · Annual Symposium on Foundations of Computer Science · doi:10.1109/focs.2016.13 · Cited by 46
Amortized entanglement of a quantum channel and approximately teleportation-simulable channels
Kaur, E., & Wilde, M. M. (2017) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aa9da7 · Cited by 48
An Introduction to Quantum Computing, without the Physics
Nannicini, G. (2020) · SIAM Review · doi:10.1137/18m1170650 · Cited by 46
More Entanglement Implies Higher Performance in Channel Discrimination Tasks
Bae, J., Chruściński, D., & Piani, M. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.140404 · Cited by 49
Metadata
- Authors
- Kitaev, A. Y.
- Corresponding author
- Kitaev, A Yu
- ORCID
- Kitaev, A Yu
- Venue
- Russian Mathematical Surveys
- Publisher
- Steklov Mathematical Institute
- Published
- 1997-12-31
- Volume
- 52
- Issue
- 6
- Pages
- 1191-1249
- ISSN
- 0036-0279 (ISSN-L)
- 1468-4829
- OpenAlex
- W2159566537
- Semantic Scholar
- de6bbbce317c9510cd0e624b7c8d66eff5bf8977
- Cited by
- 1,424
- References
- 50
- Citations by year
- 2026: 68 · 2025: 160 · 2024: 118 · 2023: 128 · 2022: 75 · 2021: 89 · 2020: 83 · 2019: 60 · 2018: 52 · 2017: 32 · 2016: 38 · 2015: 34 · 2014: 46 · 2013: 42 · 2012: 34
- FWCI
- 16.04
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Russian Mathematical Surveys (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Coding theory and cryptography
- Keywords
- quantum computing, quantum algorithms, quantum error correction, quantum error-correcting codes, measurement operators, reversible computation, quantum control, mixed states
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-06, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-06 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
