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Characterizing quantum gates via randomized benchmarking
Open accessAbstract
We describe and expand upon the scalable randomized benchmarking protocol proposed in Phys. Rev. Lett. 106, 180504 (2011) which provides a method for benchmarking quantum gates and estimating the gate dependence of the noise. The protocol allows the noise to have weak time and gate dependence, and we provide a sufficient condition for the applicability of the protocol in terms of the average variation of the noise. We discuss how state-preparation and measurement errors are taken into account and provide a complete proof of the scalability of the protocol. We establish a connection in special cases between the error rate provided by this protocol and the error strength measured using the diamond norm distance.
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TL;DR
This paper analyzes and expands a scalable randomized benchmarking protocol for Clifford gates, providing zeroth- and first-order fitting models that tolerate weak time- and gate-dependent noise and account for state-preparation and measurement errors. It proves scalability and establishes an exact relationship between the benchmarking error rate and diamond-norm error in special Pauli cases.
Key findings
- The protocol requires at most \(O(n^2)\) quantum gates, \(O(n^4)\) classical pre-processing, and a number of single-shot repetitions independent of \(n\); random Clifford sampling and decomposition total \(O(n^4)\) classical operations.
- The zeroth-order model gives exponential fidelity decay when noise is gate- and time-independent; the first-order model adds a term \(C_1(m-1)(q(m)-p^2)p^{m-2}\) that measures gate-dependence of the noise.
- State-preparation and measurement errors are absorbed into fit parameters \(A_0,B_0\) or \(A_1,B_1,C_1\); the protocol is robust against them unless the fidelity curve is constant, which occurs only for \(p=0\), \(p=1\), or \(A_0=0\).
- For generalized Pauli channels, the paper establishes an exact relationship between the benchmarking error rate \(r=1-F_{\Lambda,I}\) and the diamond norm \(\|\Lambda-I\|_\diamond\).
Methods
- For each sequence length \(m\), the protocol generates \(K_m\) sequences of \(m+1\) Clifford operations: the first \(m\) are uniformly random, and the final operation is the inverse of their composition. It measures the survival probability, averages over sequences, repeats for different \(m\), and fits the fidelity decay.
- The analysis uses a perturbative expansion of gate- and time-dependent errors about the average error operator, Clifford twirling, and zeroth- and first-order fitting models. A sufficient condition for neglecting higher orders is derived using the \(\|\cdot\|_{H^{1\to1}}\) norm.
- Scalability is proven using the symplectic representation of the Clifford group, Hoeffding’s inequality for the number of repetitions, and decomposition of Clifford operations into one- and two-qubit generators. No new experimental data are reported.
Limitations
- The protocol assumes weak time- and gate-dependence of the noise; the paper states a sufficient condition in terms of the average variation of the noise, and higher-order terms are not negligible when this condition fails.
- The paper notes a pathological failure case: if the noise is gate-dependent and equal to the exact inverse of the target gate, benchmarking can report zero error despite substantial error on each gate.
- The exact relationship between the benchmarking error rate and the diamond norm is established only for generalized Pauli channels; the paper states that a general explicit relationship is impossible and leaves broader connections for future work.
- The authors state that how the sufficient condition depends on the number of qubits \(n\) is left for further research, and that noise correlations between qubits may create large errors on Clifford elements even when errors on a generating set are controlled.
- Evident from the text: this is a theoretical and protocol-analysis paper; it does not provide new experimental validation.
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In the paper
“The average number of generators in such a decomposition is 1.875 which implies that even for the single-qubit case our protocol takes no more time to implement than that of [26].”
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“In the single-qubit case, our RB scheme can be implemented by explicitly writing down the 24 elements of the Clifford group decomposed into a sequence of the same generators that are randomly applied in [26].”
What the sources say
- Not checked yetcites [26] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Physical Review A 77 , 012307 (2008).
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“We note that for each m , in the limit of K m → ∞ , F seq ( m,ψ ) converges to the exact (uniform) average, F g ( m,ψ ), over all sequences,”
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“Moreover χ 0 , 0 for Λ k dep decreases to 1 d 2 exponentially in k.”
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“First, we look at the zeroth order fitting model F (0) g ( m, | ψ 〉 ) and note that F (0) g ( m, | ψ 〉 ) is exact in the case that the noise is independent of both the gate chosen and time, ie. Λ i j ,j = Λ.”
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“Hence the m+1'th gate is decoupled from the rest of the sequence and we have”
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“Therefore in the case of randomized benchmarking (where we define the error rate r = 1 -F Λ , I ) if Λ is a generalized Pauli channel, r and ‖ Λ -I‖ ⋄ are related by,”
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“As a simple corollary to Eq. (5.4) note that if E 1 and E 2 are depolarizing channels with fidelity parameters p 1 and p 2 respectively then,”
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“To prove Eq. (5.4) using the semidefinite program in [49] first note that Φ = E 1 -E 2 has action,”
What the sources say
- Not checked yetcites [49] J. Watrous, Theory of Computing 5 (2009). source
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“The time-complexity in decomposing a symplectic matrix into the sequence of one and two-qubit Clifford operations given above is O ( n 3 ) since one needs to solve linear systems of equations to obtain T 1 through T 5 .”
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“An unphysical but intuitively simple example is when the error is gate-dependent and equal to the exact inverse of the target gate. The error rate given by the protocol is always equal to zero however in actuality there is substantial error on each gate (see Sec. IV B).”
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In the paper
“We provide a detailed proof that our protocol requires at most O ( n 2 ) quantum gates, O ( n 4 ) cost in classical pre-processing (to select each gate-sequence), and a number of single-shot repetitions that is independent of n .”
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“Suppose the noise is timeindependent and for each i, Λ i = C † i . Then F g ( m,ψ ) = 1 for every m even though there is substantial error on each C i and so benchmarking clearly fails.”
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“First, we note that the size of the Clifford group scales as 2 O ( n 2 ) and so the number of sequences of length m scales as 2 mO ( n 2 ) . Hence if full averaging over the Clifford group is required for each sequence length, our protocol does not scale well in either of n or m .”
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“p = 1: This case corresponds to Λ being the identity channel which means all gates are perfect. Again, in practice this situation is unlikely as the implementation of any gate will have some associated error. Note that in this case the fidelity is equal to A 0 + B 0 which is just tr(Λ( ρ ψ ) E ψ )) = tr( ρ ψ E ψ ).”
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“p = 0: This occurs if and only if Λ is the totally depolarizing channel and in this case the fidelity is constant at B 0 = tr( E ψ ) d ≤ 1 d .”
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“From Eq. (3.26) an exponential decay occurs if and only if A 0 is non-zero and p lies in (0 , 1). Hence no decay occurs if and only if one of p = 0, p = 1 or A 0 = 0 occurs.”
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“Now suppose that E 2 = I in Eq. (5.4). Then, r 0 = 1 and for every 1 ≤ i ≤ d 2 -1, r i = 0.”
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“Note that the fidelity will be equal to B 0 in this case and so can take any value in [0 , 1].”
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“Note that k is explicitly independent of m and n which provides a solution to 1.”
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“As a direct comparison, performing process tomography on a 4 qubit system already requires 65536 measurements.”
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“Let S k ( m, | ψ 〉 ) = F ⃗ i m g ( m, | ψ 〉 )+ .... + F ⃗ i m g ( m, | ψ 〉 ) k be the normalized k-fold sum of the random variable F i ⃗ m g ( m, | ψ 〉 ) and note that BX [ S k ( m, | ψ 〉 )] = F g ( m,ψ ). A probablistic bound on | S k ( m, | ψ 〉 ) -F g ( m,ψ ) | is given by H¨ oeffding's inequality,”
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“Since randomly choosing 2 n elements of the Pauli group that satisfy the required commutation relations is equivalent to inductively choosing random solutions to 2 n sets of linear equations (which requires O ( n 3 ) operations), we can produce a random Clifford element in O ( n 4 ) (classical) operations.”
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“The protocol of [26] is limited to the single-qubit case and fits the observed fidelity decay averaged over sequences of single-qubit gates (where each gate consists of a random generator of the Clifford group composed with a random Pauli operator) to an exponential.”
What the sources say
- Not checked yetcites [26] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Physical Review A 77 , 012307 (2008).
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In the paper
“For instance [18] assumes the highly idealized situation of the noise being independent of the chosen gate, in which case the fidelity decay curve averaged over randomly chosen unitaries takes the form of an exponential (in the sequence length).”
What the sources say
- Not checked yetcites [18] J. Emerson, R. Alicki, and K. Zyczkowski, Journal of Optics B: Quantum and Semiclassical Optics 7 , S347 (2005). source
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“However, conditions for when the assumption of an exponential decay is valid, specifically in the realistic case of gatedependent and time-dependent noise, were not given.”
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“Our zeroth order model directly shows that for time-independent and gate-independent errors the fidelity decay is indeed modeled by an exponential decay, and the decay rate produces an estimate for the average error rate of the noise.”
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“The fitting formula shows that gatedependent errors can lead to a deviation from the exponential decay (defining a partial test for such effects in the noise), which was illustrated via numerical examples in [25].”
What the sources say
- Not checked yetcites [25] E. Magesan, J. M. Gambetta, and J. Emerson, Phys. Rev. Lett. 106 , 180504 (2011). source
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In the paper
“Thus twirling a quantum operation over the Clifford group produces a depolarizing channel and the average fidelity is invariant under the twirling operation.”
What the sources say
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“Other important shortcomings of these previous RB protocols are that extensions to multi-qubit systems are either not scalable or not well understood, and it is unclear how to explicitly account for state preparation and measurement errors.”
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“In this paper we give a full analysis of the scalable multi-qubit randomized benchmarking protocol for Clifford gates we proposed in [25] which overcomes the shortcomings described above.”
What the sources say
- Not checked yetcites [25] E. Magesan, J. M. Gambetta, and J. Emerson, Phys. Rev. Lett. 106 , 180504 (2011). source
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“The Monte-Carlo methods also have the drawback of assuming negligible state-preparation and measurement errors.”
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“One can show that r and r P are related via r P = ( d +1) r d .”
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In the paper
“The key point to note here is that the noise is highly dependent on the gate chosen and so we expect that the sufficient condition derived above for ignoring higher order terms will not be satisfied (ie. γ in this example will be far from 0).”
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In the paper
“The difference between Eq.'s (3.44) and (3.26) is the C 1 ( m -1)( q ( m ) -p 2 ) p m -2 term contained in Eq. (3.44), which can be thought of as a measure of the gate-dependence of the noise.”
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“The advantages of these methods are that the average fidelity of each gate can be estimated and the scheme is efficient in n .”
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In the paper
“The diamond norm ‖ · ‖ ⋄ is a candidate however by Eq. (2.11) ‖ ‖ H 1 → 1 is much weaker than ‖ · ‖ ⋄ . Therefore γ associated with ‖ ‖ H 1 → 1 will be much smaller than γ associated with ‖ · ‖ ⋄ , providing a tighter bound on the fidelity difference.”
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“In general an explicit relationship will be impossible to obtain, however we show that in certain cases that are relevant in various fault-tolerant noise models we can obtain such a relationship.”
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In the paper
“Thus the protocol is robust against any state preparation or measurement errors unless these errors create a constant fidelity curve.”
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In the paper
“This discussion shows that a constant fidelity decay curve can only occur in extreme cases and so it is safe to assume the protocol is independent of state preparation and measurement errors.”
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- Rests on an assumption: In practice, gate operations are reasonably precise, so substantial state-preparation or measurement errors are unlikely.
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In the paper
“While this number is large it is independent of n and thus compares favourably with quantum process tomography which scales as 16 n .”
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- The authors state this as an assumption: The physical gates have errors on the order of 10^-4.
- The authors state this as an assumption: The accuracy ε can be relaxed as m grows large.
- The authors state this as an assumption: The range b-a can be assumed relatively small for small m but converges to 1 - 1/d as m grows large.
- The authors state this as an assumption: m=100 and the fidelity decay curve is well-approximated by an exponential.
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In the paper
“Second, the unitary group can be generated by adding just one single-qubit rotation not in the Clifford group (for instance the π 8 -gate).”
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In the paper
“It is not difficult to see that benchmarking the Clifford group provides significant information for both fault-tolerant quantum computation as well as obtaining a benchmark for a generating set of the full unitary group.”
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- Rests on an assumption: Any realistic implementation of a quantum computer will have to take advantage of error-correction codes to perform fault-tolerant quantum computation.
- Rests on an assumption: Most of the codes used in fault-tolerant theory are stabilizer codes.
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In the paper
“As previously noted, the noise associated to a multi-qubit Clifford element is given by the noise associated to the sequence of generators comprising the Clifford.”
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“The fact that most of the codes used in fault-tolerant theory are stabilizer codes implies that the encoding and decoding operations that have to be performed can be chosen to be Clifford operations.”
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- Rests on an assumption: Most of the codes used in fault-tolerant theory are stabilizer codes.
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“Rigorous fault-tolerant analyses sometimes invoke the diamond norm as a measure of the error strength rather than the weaker characterization provided by the average fidelity.”
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In the paper
“While benchmarking the full unitary group would be ideal, this is a provably inefficient task since just generating a Haar-random unitary operator is inefficient in n . On the other hand as we have shown here benchmarking the Clifford group is an efficient task.”
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“From the expression given in Eq. (2.2) one can see that the diamond norm is essentially a 'worst-case' maximization over input (entangled) states.”
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In the paper
“In this case the noise must have a strong gate-dependence because otherwise q -p 2 would be small which implies the two fitting functions would produce similar estimates for p .”
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In the paper
“The protocol allows for time and gatedependent noise and the fitting models for the fidelity function take into account state preparation and measurement errors.”
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In the paper
“Additionally, if one were able to obtain an estimate of the minimum gate fidelity from knowledge of the average fidelity they could use the direct relationship between the minimum gate fidelity and diamond norm given by Eq. (2.21) to obtain information about the error strength in terms of the diamond norm.”
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In the paper
“A result that may be useful in this direction of research is the 'concentration of measure effect' of the gate fidelity which implies that as n increases, the measure of the set of states which produce a fidelity close to the minimum yet far from the average is exponentially small in n [41, 42].”
What the sources say
- Not checked yetcites [41] E. Magesan, R. Blume-Kohout, and J. Emerson, Phys. Rev. A 84 , 012309 (2011).
- Not checked yetcites [42] E. Magesan, Quant. Inf. Comp. 11 , 0466 (2011). source
Strength:hedged· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
The authors state this as an assumption.
The actual implementation of each Clifford gate C_i at time j can be represented as E_i,j = Λ_i,j ◦ C_i for some error map Λ_i,j.
“Suppose that the actual implementation of C i at time j (1 ≤ j ≤ M ) results in the map E i,j with E i,j = Λ i,j ◦ C i for some error map Λ i,j .”
The authors state this as an assumption.
Each operation C_i_j is allowed to have some error, represented by Λ_i_j,j.
“By assumption each operation C i j is allowed to have some error, represented by Λ i j ,j , and each sequence can be modelled by the operation,”
The authors state this as an assumption.
In the simplest ideal scenario, the noise operator at each step is independent of the applied gate and time-invariant.
“assuming the simplest (ideal) scenario where the noise operator at each step is independent of the applied gate (and is also time-invariant)”
The authors state this as an assumption.
The perturbation δΛ_i,j is small relative to Λ.
“Our approach will be valid provided δ Λ i,j is a small perturbation from Λ in a sense to be made precise later.”
The authors state this as an assumption.
The gate-dependence of the noise is weak enough that the zeroth-order expression in Eq. (3.26) is a valid model for the fidelity decay curve.
“For simplicity of the discussion let us assume the gate-dependence of the noise is weak enough so that the zeroth order expression given in Eq. (3.26) is a valid model for the fidelity decay curve.”
The authors state this as an assumption.
The physical gates have errors on the order of 10^-4.
“Let us assume that the physical gates have errors on the order of 10 -4 .”
The authors state this as an assumption.
The accuracy ε can be relaxed as m grows large.
“Intuitively, since the fidelity curve decays in sequence length it is reasonable to assume that ǫ can be relaxed as m grows large.”
The authors state this as an assumption.
The range b-a can be assumed relatively small for small m but converges to 1 - 1/d as m grows large.
“Similarly, b -a can be assumed to be relatively small for small values of m but will converge to 1 -1 d as m grows large.”
The authors state this as an assumption.
m=100 and the fidelity decay curve is well-approximated by an exponential.
“Let us assume m = 100 and a fidelity decay curve that is well-approximated by an exponential.”
The paper relies on this without support.
In practice, gate operations are reasonably precise, so substantial state-preparation or measurement errors are unlikely.
“Since gates are reasonably precise in practice, this situation occurs when at least one of state preparation or measurement has substantial error.”
The paper relies on this without support.
In quantum computation, the measure of accessible states (states reachable in polynomial time using a generating set for the unitary group) is equal to 0.
“In quantum computation it is the case that the measure of accessible states (states that can be reached in polynomial time using a generating set for the unitary group) is equal to 0.”
The paper relies on this without support.
Any realistic implementation of a quantum computer will have to take advantage of error-correction codes to perform fault-tolerant quantum computation.
“First, any realistic implementation of a quantum computer will have to take advantage of error-correction codes in order to perform fault-tolerant quantum computation.”
The paper relies on this without support.
Most of the codes used in fault-tolerant theory are stabilizer codes.
“The fact that most of the codes used in fault-tolerant theory are stabilizer codes implies that the encoding and decoding operations that have to be performed can be chosen to be Clifford operations.”
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?
The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Existing works have already proposed to adopt randomly generated benchmarks for evaluating the error rates of diferent gate operations regarding a particular NISQ device [83, 84, 106].”
From QASMBench: A Low-Level Quantum Benchmark Suite for NISQ Evaluation and Simulation (2023) · cites this paper as [84]
Experimental — accuracy not yet measuredIn the citing paper
“In these noisy intermediate-scale quantum (NISQ) systems [4], performance of isolated gates may not predict the behavior of the system.”
From Validating quantum computers using randomized model circuits (2019) · cites this paper as [4]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Conflicting time complexities for Clifford decomposition
MinorThe paper says it describes a method that decomposes a Clifford element in O(n2) time, but later assigns O(n3) time to decomposing it into the described sequence of operations.
“Any Clifford element can be decomposed into a sequence of O ( n 2 ) one and two-qubit generators in O ( n 2 ) time [52] (alternatively, there are slower methods which produce a 'canonical' decomposition into O ( n 2 / log n ) generators [53]). We describe this method which again utilizes the symplectic representation of the Clifford group.”
“The time-complexity in decomposing a symplectic matrix into the sequence of one and two-qubit Clifford operations given above is O ( n 3 )”
Inconsistent number of operators in the generalized Pauli basis
MinorIndexing the proposed basis from 0 through d2 gives d2+1 operators. Elsewhere the paper indexes such a basis from 0 through d2−1, giving d2 operators.
“Suppose E 1 and E 2 are Pauli channels, or more generally any channels with Kraus operators given by an orthogonal (normalized to d) basis of unitary operators { P i } d 2 i =0 (which we call generalized Pauli channels),”
“Noting that {| ψ i 〉 := P i ⊗ BD | ψ 0 〉} d 2 -1 i =0 forms an orthonormal basis of maximally entangled states for C d ⊗ C d”
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Metadata
- Authors
- Magesan, E., Gambetta, J. M., & Emerson, J.
- Affiliations
- University of Waterloo (CA)
- IBM Research - Thomas J. Watson Research Center (US)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2012-04-11
- Volume
- 85
- Issue
- 4
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2017477211
- Cited by
- 464
- References
- 50
- Citations by year
- 2026: 33 · 2025: 67 · 2024: 48 · 2023: 50 · 2022: 47 · 2021: 54 · 2020: 42 · 2019: 35 · 2018: 15 · 2017: 19 · 2016: 15 · 2015: 16 · 2014: 14 · 2013: 5 · 2012: 4
- FWCI
- 12.19
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- arXiv (Cornell University)
- 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 › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- randomized benchmarking, quantum gate characterization, gate noise, scalability, state preparation and measurement errors, quantum error rates
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