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
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Claims the paper makes
86 No basis given · 26 Own interpretation · 12 Not checked yet · 5 Own result · 2 Own method
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“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.
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“The norm of an admissible transformation is always equal to one.”
§ Quantum computations: algorithms and error correction > 8 below).
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“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.
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“Corollary. C^ is a complete basis for reversible computations.”
§ Quantum computations: algorithms and error correction > Corollary. C^ is a complete basis for reversible computations.
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“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
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“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.
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“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
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- 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.
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“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.
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“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
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- 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.
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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
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- Probably not a substantial claim. 19
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“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.
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“Quantum probability is additive, just as classical probability is.”
§ Quantum computations: algorithms and error correction > 3.1. Basic notions and notation.
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“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].
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“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.
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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]
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- Also stated in: § §1. Abelian problem on the stabilizer · plain
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“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.
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- Also stated in: § 3.2. Transformations of mixed states. · plain
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“First, any mixed state ρ £ D(N) can be represented as”
§ Quantum computations: algorithms and error correction > 3.1. Basic notions and notation.
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“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
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- 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
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“Unfortunately, Λ(5) can only be represented approximately, which gives rise to some technical inconvenience.”
§ Quantum computations: algorithms and error correction > {|°^ * a = ° (28)
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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
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“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.
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“The information operators are defined to within multiplication by a verification operator.”
§ Quantum computations: algorithms and error correction > 9.1. Algebraic preparation.
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“Besides, each element of Q' can be represented in the basis Q.”
§ Quantum computations: algorithms and error correction > (29) v2
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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.
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“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 :
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“(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.)
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“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.
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“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.
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“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.
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“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.
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“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.
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“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\.
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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]
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“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.
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- 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.
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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.
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- 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.
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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
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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.
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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),
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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.
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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.
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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.
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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.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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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.
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It may state two separate claims as one.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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Elementary gates for quantum computation
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Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
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- 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
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Quantum Formulas: A Lower Bound and Simulation
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Upper bounds on the noise threshold for fault-tolerant quantum computing
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Constructing arbitrary single-qubit fault-tolerant gatesPreprint
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Quantum Hall Effect and Hierarchy of Planck ConstantsPreprint
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Chernoff Bound for Quantum Operations is FaithfulPreprint
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Magic state distillation with punctured polar codesPreprint
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Bipartite Quantum Interactions: Entangling and Information Processing Abilities
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Zero-knowledge against quantum attacksPreprint
Watrous, J. (2005) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0511020 · Cited by 6
Quantum Logspace Algorithm for Powering Matrices with Bounded NormPreprint
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Supercomputer simulations of transmon quantum computers
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Silicide-based Josephson field effect transistors for superconducting qubits
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Application of zero-noise-extrapolation-based quantum error mitigation to a silicon spin qubit
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Quantum Snakes on a Plane: Mobile, Low-Dimensional Logical Qubits on a 2D Surface
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Toward quantum computing gauge theories of nature
Davoudi, Z. (2026) · The European Physical Journal Special Topics · doi:10.1140/epjs/s11734-026-02313-y · Cited by 5
First-order crosstalk mitigation in parallel quantum gates driven with multi-photon transitions
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Neural-network-designed three-qubit gates robust against charge noise and crosstalk in silicon
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Going beyond gadgets: the importance of scalability for analogue quantum simulators
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Shadow Simulation of Quantum Processes
Zhao, X., Wang, X., & Chiribella, G. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.120804 · Cited by 6
Adaptive denoising quantum state preparation in a dynamic environment
Wang, Z.-M., & Chen, T.-Z. (2024) · Physical Review Research · doi:10.1103/physrevresearch.6.043195 · Cited by 5
Error-correctable efficient quantum homomorphic encryption using Calderbank–Shor–Steane codes
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Exact spectral form factors of noninteracting fermions with Dyson statistics
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Meta-learning assisted robust control of universal quantum gates with uncertainties
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Tightly-integrated quantum–classical computing using the QHDL hardware description language
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Continuous-variable fault-tolerant quantum computation under general noise
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Computable lower bounds on the entanglement cost of quantum channels
Lami, L., & Regula, B. (2022) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aca731 · Cited by 5
Advances in quantum error correction based on superconducting quantum systems
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Decoding topological XYZ 2 codes with reinforcement learning based on attention mechanisms
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Feynman Meets Turing: The Uncomputability of Quantum Gate-Circuit Emulation and Concatenation
Boche, H., Bock, Y. N., del Toro, Z. G., & Fitzek, F. H. P. (2024) · IEEE Transactions on Computers · doi:10.1109/tc.2024.3506861 · Cited by 5
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Barg, A., Coble, N. J., Hangleiter, D., & Kang, C. (2025) · IEEE Transactions on Information Theory · doi:10.1109/tit.2025.3592631 · Cited by 9
Exploring boundaries of quantum convex structures: Special role of unitary processes
Puchała, Z., Jenčová, A., Sedlák, M., & Ziman, M. (2015) · Physical Review A · doi:10.1103/physreva.92.012304 · Cited by 5
Probability in quantum computation and quantum computational logics: a survey
Chiara, M. L. D., Giuntini, R., & Sergioli, G. (2014) · Mathematical Structures in Computer Science · doi:10.1017/s0960129512000734 · Cited by 5
Fesenko, A. (2014) · Cybernetics and Systems Analysis · doi:10.1007/s10559-014-9672-y · Cited by 5
Instability of nondiscrete free subgroups in lie groups
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Approximate Quantum Fourier Transform and Quantum Algorithm for Phase Estimation
Prokopenya, A. N. (2015) · Lecture notes in computer science · doi:10.1007/978-3-319-24021-3_29 · Cited by 5
Global Estimates of Errors in Quantum Computation by the Feynman–Vernon Formalism
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Метод Кротова в задачах оптимального управления замкнутыми квантовыми системами
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Gülbahar, B. (2020) · Scientific Reports · doi:10.1038/s41598-020-67364-0 · Cited by 7
Surface code for low-density qubit array
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Dimension Reduction and Redundancy Removal through Successive Schmidt Decompositions
Daskin, A., Gupta, R., & Kais, S. (2023) · Applied Sciences · doi:10.3390/app13053172 · Cited by 5
Quantum Wasserstein distance of order 1 between channels
Duvenhage, R., & Mapaya, M. (2023) · Infinite Dimensional Analysis Quantum Probability and Related Topics · doi:10.1142/s0219025723500066 · Cited by 6
Decoherence limiting the cost to simulate an anharmonic oscillator
Propp, T. B., Ray, S., DeBrota, J. B., Albash, T., & Deutsch, I. (2023) · Physical Review A · doi:10.1103/physreva.108.062219 · Cited by 6
Guest Column: The 7 faces of quantum NP
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Spatial quantum error correction threshold
Mizel, A. (2024) · Physical Review A · doi:10.1103/physreva.109.022406 · Cited by 16
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The Non-m-Positive Dimension of a Positive Linear Map
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Self-consistent quantum tomography with regularization
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Korolev, S. B., & Golubeva, T. Y. (2022) · Physics Letters A · doi:10.1016/j.physleta.2022.128149 · Cited by 5
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An Algorithm for Computing a Basis of a Finite Abelian Group
Karagiorgos, G., & Poulakis, D. (2011) · Lecture notes in computer science · doi:10.1007/978-3-642-21493-6_11 · Cited by 5
Quantum Codes From Toric Surfaces
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Simple proof of confidentiality for private quantum channels in noisy environments
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Thoughts on Noise and Quantum ComputationPreprint
Kalai, G. (2005) · arXiv (Cornell University) · doi:10.48550/arxiv.quant-ph/0508095 · Cited by 5
Sampling and the complexity of naturePreprint
Hangleiter, D. (2020) · arXiv (Cornell University) · doi:10.17169/refubium-28790 · Cited by 8
Synthesis of Quantum Circuits with an Island Genetic AlgorithmPreprint
Miranda, F. T., Balbi, P. P., & Costa, P. C. S. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2106.03115 · Cited by 5
Approximation, Proof Systems, and Correlations in a Quantum WorldPreprint
Gharibian, S. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1301.2632 · Cited by 6
Limitations in quantum computing from resource constraintsPreprint
Fellous-Asiani, M., Chai, J. H., Whitney, R. S., Auffèves, A., & Ng, H. K. (2020) · arXiv (Cornell University) · doi:10.1103/prxquantum.2.040335 · Cited by 4
A Substrate Scheduler for Compiling Arbitrary Fault-Tolerant Graph States
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Takeuchi, Y. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.050601 · Cited by 5
Transforming Water Quality Assessment: A Quantum AI Approach
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Quantum Circuit Complexity as a Physical Observable
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A Survey: SWAP Test and Its Applications to Quantum Complexity Theory
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Noisy probabilistic error cancellation and generalized physical implementability
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Introduction to quantum error correction with stabilizer codes
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Ladder Operator Block-Encoding
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Undecidability in spacetime geometry via the AdS/CFT correspondence
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A Review of Formal Methods in Quantum-Circuit Verification
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Advancing quantum machine learning from conceptual design to practical use
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Logical Majorana Fermions for Fault-Tolerant Quantum Simulation.
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Execution of a quantum algorithm for calculating energies of some quantum mechanical systems
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Benchmarking of universal qutrit gates
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Discrete-outcome sensor networks. II. Multiple detection events and grouping detectors
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Single-qubit rotation algorithm with logarithmic Toffoli count and gate depth
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Short two-qubit pulse sequences for exchange-only spin qubits in two-dimensional layouts
Chadwick, J. D., Guerreschi, G. G., Luthi, F., Mądzik, M. T., Mohiyaddin, F. A., Prabhu, P., Schmitz, A. T., Litteken, A., Premaratne, S., Bishop, N. C., et al. (2025) · Physical Review A · doi:10.1103/physreva.111.052616 · Cited by 6
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
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