Scalable noise estimation with random unitary operators
Open accessAbstract
We describe a scalable stochastic method for the experimental measurement of generalized fidelities characterizing the accuracy of the implementation of a coherent quantum transformation. The method is based on the motion reversal of random unitary operators. In the simplest case our method enables direct estimation of the average gate fidelity. The more general fidelities are characterized by a universal exponential rate of fidelity loss. In all cases the measurable fidelity decrease is directly related to the strength of the noise affecting the implementation, quantified by the trace of the superoperator describing the non-unitary dynamics. While the scalability of our stochastic protocol makes it most relevant in large Hilbert spaces (when quantum process tomography is infeasible), our method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device. By varying over different experimental arrangements and error-correction strategies, additional information about the noise can be determined.
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TL;DR
The paper proposes a scalable stochastic protocol for estimating noise in quantum implementations by applying random unitary operators and their inverses (motion reversal), enabling direct estimation of average gate fidelity and generalized fidelities. It shows the measurable fidelity loss depends on the trace of the cumulative noise superoperator and can be estimated with accuracy \(O(1/\sqrt{DN})\) for \(N\) independent experiments, avoiding the exponential scaling of quantum process tomography.
Key findings
- Under Haar-random motion reversal, the average gate fidelity depends only on a single noise-strength parameter \(p \in [0,1]\), equivalently \(\mathrm{Tr}(\hat{\Lambda})\), and can be read out as the residual population of the initial computational basis state.
- The method estimates noise strength with accuracy \(O(1/\sqrt{DN})\) for \(N\) independent experiments, where \(D\) is the Hilbert-space dimension. In contrast, quantum process tomography requires \(O(D^4)\) experiments and manipulation of \(D^2 \times D^2\) matrices; Nielsen’s direct \(F_g\) method requires \(D^4\) experiments but only \(D \times D\) matrices.
- Generalized iterated motion-reversal and Loschmidt-echo sequences yield a universal exponential decay of averaged fidelity depending only on noise strength; as the number of iterations \(n\) grows, \(F_n \to D^{-1}\).
- For continuous-time weak Markovian noise, under an ergodic hypothesis and small noise parameter \(\epsilon\), the paper derives a universal exponential fidelity decay. A generalized Loschmidt echo with noise in both forward and backward sequences is conjectured to satisfy relation (31), with analytic and numerical evidence for unitary noise.
Methods
- Theoretical/analytical protocol: apply a Haar-random unitary \(U\) and its inverse \(U^{-1}\) to an initial state, average over the Haar measure, and use superoperator representation. The Haar-averaged superoperator is \(U(D)\)-invariant and equivalent to a depolarizing channel characterized by \(\mathrm{Tr}(\hat{\Lambda})\).
- Generalized discrete-time analysis: iterated motion reversal and Loschmidt-echo sequences with independent random unitaries, arbitrary mixed initial states, and averaging over Haar measure.
- Continuous-time analysis: Markovian master equation with controlled reversible dynamics and a small noise parameter \(\epsilon\), using a cumulant expansion and an ergodic hypothesis for the time-dependent generator.
Limitations
- The paper states the only inefficiency is the requirement to experimentally implement a Haar-random unitary: its decomposition into elementary one- and two-qubit gates requires an exponentially long gate sequence. It leaves open whether efficient random circuits can provide adequate randomization.
- The paper states it assumes the noise superoperator is independent of the specific target transformation, but notes that actual noise generally depends sensitively on the target \(U\) and the gate sequence; it argues this assumption may become statistically well motivated but does not prove it.
- The generalized Loschmidt-echo relation (31) is not directly evaluated analytically in the general case; the paper reports only analytic and numerical evidence for unitary noise and conjectures it holds under general noise. Continuous-time results rely on the stated ergodic hypothesis and small-\(\epsilon\), long-time conditions.
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Claims the paper makes
0 of 12 cited sources checked · 0 supported · 0 flagged
- Findingnone given§ Scalable Noise Estimation with Random Unitary Operators · p. 1
The more general fidelities are characterized by a universal exponential rate of fidelity loss.
“The more general fidelities are characterized by a universal exponential rate of fidelity loss.”
Quote in the paperNot a substantial claim - Findingnone given§ Scalable Noise Estimation with Random Unitary Operators · p. 1
In all cases the measurable fidelity decrease is directly related to the strength of the noise affecting the implementation, quantified by the trace of the superoperator describing the non-unitary dynamics.
“In all cases the measurable fidelity decrease is directly related to the strength of the noise affecting the implementation - quantified by the trace of the superoperator describing the non-unitary dynamics.”
Quote in the paperNot a substantial claim - Findingreasoning§ GENERALIZED FIDELITIES IN A DISCRETE-TIME SCENARIO · p. 4
Assuming the cumulative noise for each step has the same strength p_j = p, one obtains for arbitrary noise a universal exponential form for the decay of fidelity in the Loschmidt echo sequence.
“As before, we can simplify this result by assuming that the cumulative noise for each step has the same strength ( p j = p ), in which case we obtain for arbitrary noise a universal exponential form for the decay of fidelity”
Quote in the paperSubstantial claim- The authors state this as an assumption: The cumulative noise for each U_j, or each step, has the same strength p_j = p.
- Findingcitation§ GENERALIZED FIDELITIES IN A DISCRETE-TIME SCENARIO · p. 4
In the limit of large n, F_n approaches D^{-1}, as may be expected from the average fidelity between random states.
“In the limit of large n , we see that F n → D -1 , as may be expected from the average fidelity between random states [16].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [16] source
Not checked in the free run.
- Findingreasoning§ GENERALIZED FIDELITIES IN A DISCRETE-TIME SCENARIO · p. 4
For an iterated motion reversal sequence, the Haar-averaged noise operator is a depolarizing channel with a strength parameter.
“As noted above and shown in the Appendix, ˆ Λ ave ≡ ∫ dU ˆ U ˆ Λ ˆ U -1 is a depolarizing channel with strength parameter”
Quote in the paperSubstantial claim
- Findingnone given§ APPENDIX: HAAR AVERAGED SUPEROPERATORS · p. 7
The Haar-averaged superoperator corresponding to the noise under the imperfect motion-reversal protocol, averaged over all possible unitary operators, is a U(D)-invariant superoperator.
“The Haar-averaged superoperator corresponding to the noise under the imperfect motion-reversal protocol, averaged over all possible unitary operators, is a U ( D ) -invariant superoperator”
Quote in the paperSubstantial claim - Findingreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 2
The average gate fidelity provides an indicator that is independent of the choice of initial state.
“The measure dψ denotes the natural, unitarily invariant (Fubini-Study) measure on the set of pure states and hence the average gate fidelity provides an indicator that is independent of the choice of initial state.”
Quote in the paperSubstantial claim - Findingreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
After the motion reversal sequence has been applied experimentally, the single parameter p characterizing the average gate fidelity appears as the residual population of the initial state.
“The protocol is now clear: after the motion reversal sequence has been applied experimentally, the single parameter p characterizing the average gate fidelity appears as the residual population of the initial state.”
Quote in the paperSubstantial claim - Findingown data§ GENERALIZED FIDELITIES IN A DISCRETE-TIME SCENARIO · p. 5
For the generalized Loschmidt echo with noise in both forward and backward sequences, for the special case of unitary noise there is analytic and numerical evidence supporting the conjectured relation for large n.
“While we have not directly evaluated the average of this fidelity analytically in the general case, for the special case of unitary noise we have analytic and numerical evidence supporting the relation”
Quote in the paperSubstantial claim - Findingreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
The gate fidelity for the Haar-averaged operator resulting from a motion reversal experiment depends only on the single parameter Tr(Λ), which represents the intrinsic strength of the cumulative noise.
“Hence the gate fidelity for the Haar-averaged operator resulting from a motion reversal experiment depends only on the single parameter Tr( ˆ Λ) which represents the intrinsic strength of the cumulative noise.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The noise superoperator Λ is independent of the specific target transformation but depends only on the duration of the experiment.
- Findingnone given§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
If in each of N repetitions of the motion-reversal experiment an independent random unitary is applied, then the observed average will approach the Haar-average as O(1/√(DN)).
“If in each of N repetitions of the motion-reversal experiment an independent random unitary is applied, then the observed average will approach the Haar-average as O (1 / √ DN ) .”
Quote in the paperNot a substantial claim - Findingreasoning§ GENERALIZED FIDELITIES FOR CONTINUOUS-TIME WEAK NOISE · p. 6
Replacing Γ(t) by exp(ε L_av t) and using the explicit expression yields the universal exponential decay of the fidelity.
“Then replacing ˆ Γ( t ) by exp( ǫ ˆ L av t ) and using the explicit expression (56,57) we obtain the universal exponential decay of the fidelity”
Quote in the paperSubstantial claim- The authors state this as an assumption: The ergodic mean exists and is equal to the Haar average.
- The authors state this as an assumption: The fluctuations δL(t) around the ergodic mean are normal for long t.
- Findingreasoning§ GENERALIZED FIDELITIES IN A DISCRETE-TIME SCENARIO · p. 4
Assuming the cumulative noise for each U_j has the same strength p_j = p, one obtains for arbitrary noise a universal exponential decay of the averaged fidelity depending only on the noise strength.
“Because each U j is random, we can further simplify this result by assuming that the cumulative noise for each U j has the same strength p j = p , in which case we obtain for arbitrary noise a universal exponential decay of the averaged fidelity”
Quote in the paperSubstantial claim- The authors state this as an assumption: The cumulative noise for each U_j, or each step, has the same strength p_j = p.
- Findingreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
The result that the Haar-averaged gate fidelity depends only on Tr(Λ) holds for general (possibly non-unital) noise.
“We remark that this result holds for general (possibly non-unital) noise.”
Quote in the paperSubstantial claim - Findingnone given§ INTRODUCTION · p. 1
The average gate fidelity, and some more generalized fidelities, can be estimated directly with an accuracy O(1/√(DN)) where N is the number of independent experiments.
“Specifically we show that the average gate fidelity (1), and some more generalized fidelities described below, can be estimated directly with an accuracy O (1 / √ DN ) where N is the number of independent experiments.”
Quote in the paperSubstantial claim - Interpretationnone given§ Scalable Noise Estimation with Random Unitary Operators · p. 1
In the simplest case the described method enables direct estimation of the average gate fidelity.
“In the simplest case our method enables direct estimation of the average gate fidelity.”
Quote in the paperSubstantial claim - Interpretationnone given§ Scalable Noise Estimation with Random Unitary Operators · p. 1
The scalability of the stochastic protocol makes it most relevant in large Hilbert spaces when quantum process tomography is infeasible.
“While the scalability of our stochastic protocol makes it most relevant in large Hilbert spaces (when quantum process tomography is infeasible)”
Quote in the paperSubstantial claim - Premisenone given§ INTRODUCTION · p. 1
The QPT approach to noise estimation suffers from the deficiency that the noise superoperator determined from QPT depends on the symmetries between the noise mechanisms and the choice of target transformation.
“First, often the intrinsic properties of the noise operators are of interest, but the noise superoperator determined from QPT will depend on the symmetries between the noise mechanisms and the choice of target transformation.”
Quote in the paperSubstantial claim - Premisereasoning§ INTRODUCTION · p. 1
The number of experiments that must be carried out in QPT grows exponentially in the number of qubits D^4 = 2^{4 n_q}.
“Second, the number of experiments that must be carried out grows exponentially in the number of qubits D 4 = 2 4 n q .”
Quote in the paperSubstantial claim - Premisecitation§ DISCUSSION · p. 6
From studies of quantum chaos it is known that efficient chaotic quantum maps are faithful to the universal Haar-averaged fidelity decay under imperfect motion-reversal.
“and from studies of quantum chaos it is known that efficient chaotic quantum maps are faithful to the universal Haaraveraged fidelity decay under imperfect motion-reversal [14].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [14] source
Not checked in the free run.
- Interpretationreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 2
The universal form allows evaluation of the average fidelity for more generalized motion reversal protocols.
“This universal form will allow us to evaluate the average fidelity for more generalized motion reversal protocols.”
Quote in the paperSubstantial claim - Interpretationreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 2
Under the motion reversal transformation, the CP map Λ in the gate fidelity can be interpreted as the decoherence and experimental control errors arising under the imperfect implementation of the motion reversal experiment, rather than as the noise associated with only the forward transformation U.
“Under this transformation, the CP map Λ in the gate fidelity (1) can be interpreted as the decoherence and experimental control errors arising under the imperfect implementation of the motion reversal experiment”
Quote in the paperSubstantial claim - Interpretationreasoning§ INTRODUCTION · p. 1
The overall noise strength and the associated accuracy of an implementation may be estimated by a scalable experimental method.
“We show below that the overall noise strength and the associated accuracy of an implementation may be estimated by a scalable experimental method.”
Quote in the paperSubstantial claim - Interpretationreasoning§ INTRODUCTION · p. 1
By varying over different experimental methods and noise-reduction algorithms and directly measuring the variation in the associated fidelity, the method enables estimation of more detailed characteristics of the noise.
“By varying over different experimental methods and noise-reduction algorithms and then directly measuring the variation in the associated fidelity this method enables estimation of more detailed characteristics of the noise.”
Quote in the paperSubstantial claim - Interpretationreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
The gate fidelity can be determined directly from a standard readout (projective measurement) of the final state in the computational basis.
“Hence the gate fidelity can be determined directly from a standard readout (projective measurement) of the final state in the computational basis.”
Quote in the paperSubstantial claim - Premisecitation§ DISCUSSION · p. 6
The only inefficiency of the protocol is the requirement of experimentally implementing a Haar-random unitary, whose decomposition into elementary one- and two-qubit gates requires an exponentially long gate sequence.
“The only inefficiency of our protocol is the requirement of experimentally implementing a Haar-random unitary: the decomposition into elementary one and two qubit gates requires an exponentially long gate sequence [12].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] Poźniak M, 1998, J. Phys. A: Math. Gen. source
Not checked in the free run.
- Interpretationcitation§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 2
Choosing the target transformation U randomly from the Haar measure leads to a universal form of the gate fidelity depending only on the intrinsic strength of the cumulative noise.
“The key idea is to choose the target transformation U randomly from the Haar measure [12]. This will earn us the advantage of the concentration of measure in large Hilbert spaces, as described further below, and leads to a universal form of the gate fidelity depending only on the intrinsic strength of the cumulative noise.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] Poźniak M, 1998, J. Phys. A: Math. Gen. source
Not checked in the free run.
- Interpretationreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
When the noise strength is non-negligible, an accurate estimate of p is possible with only a few experimental trials.
“When the noise strength is actually non-negligible (e.g., the noise strength does not decrease as a polynomial function of 1 /D ) an accurate estimate of p is possible with only a few experimental trials.”
Quote in the paperSubstantial claim - Premisecitation§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
The typical fluctuation for a random initial state |ψ>, given a fixed U and Λ, decreases exponentially with the number of qubits.
“As will be shown in Ref. [13], the typical fluctuation for a random initial state | ψ 〉 , given a fixed U and Λ , decreases exponentially with the number of qubits,”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [13] Blume-Kohout R Emerson J, 2005
Not checked in the free run.
- Interpretationreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
The fidelity under motion reversal of a single random U and arbitrary non-random initial state is exponentially close to the Haar-averaged fidelity.
“Hence the fidelity under motion reversal of a single random U and arbitrary (non-random) initial state is exponentially close to the Haar-averaged fidelity”
Quote in the paperSubstantial claim - Interpretationreasoning§ INTRODUCTION · p. 1
The method provides a solution to the problem of efficiently measuring which member of a set of experimental configurations and algorithmic techniques produces the most accurate implementation of an arbitrary target transformation.
“This method provides a solution to the important problem of efficiently measuring which member of a set of experimental configurations and algorithmic techniques produces the most accurate implementation of an arbitrary target transformation.”
Quote in the paperSubstantial claim - Premisecitation§ GENERALIZED FIDELITIES FOR CONTINUOUS-TIME WEAK NOISE · p. 5
The system is described by the Markovian Master Equation.
“We describe our system by the Markovian Master Equation [18, 19]”
Quote in the paperSubstantial claim0 of 2 sources checked - Premisecitation§ DISCUSSION · p. 6
Experimental results suggest that even a structured transformation such as the quantum Fourier transform is sufficiently complex to approximately average the cumulative noise to an effective depolarizing channel.
“Indeed the experimental results of Ref. [2] suggest that even a structured transformation such as the quantum Fourier transform is sufficiently complex to approximately average the cumulative noise to an effective depolarizing channel”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [2] source
Not checked in the free run.
- Interpretationreasoning§ DISCUSSION · p. 6
Because the target transformation is a Haar-random unitary, the cumulative noise measured by this method will not be biased by any special symmetries of the target transformation.
“Moreover, because the target transformation is a Haar-random unitary, the cumulative noise measured by this method will not be biased by any special symmetries of the target transformation.”
Quote in the paperSubstantial claim - Premisenone given§ INTRODUCTION · p. 1
Conventional numerical analysis of tomographic data requires manipulation of matrices of exponentially increasing dimension (D^2 × D^2).
“Third, conventional numerical analysis of the tomographic data requires the manipulation of matrices of exponentially increasing dimension ( D 2 × D 2 ).”
Quote in the paperSubstantial claim - Interpretationreasoning§ DISCUSSION · p. 6
Generalized Haar-averaged fidelities may be directly estimated with only a few experimental measurements.
“We have described how generalized Haar-averaged fidelities may be directly estimated with only a few experimental measurements.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The noise superoperator Λ is independent of the specific target transformation but depends only on the duration of the experiment.
- Interpretationreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 3
By implementing a single target transformation U randomly drawn from the Haar measure, the motion reversal gate fidelity for the single random U is exponentially close to the Haar-averaged motion reversal gate fidelity.
“By implementing a single target transformation U that is randomly drawn from the Haar measure, we gain the advantage of the concentration of measure in large Hilbert spaces: the motion reversal (gate) fidelity for the single random U is exponentially close to the Haar-averaged motion reversal (gate) fidelity.”
Quote in the paperSubstantial claim - Interpretationreasoning§ DISCUSSION · p. 6
By implementing a motion reversal sequence with a Haar-random unitary transformation, the observed fidelity decay provides a direct experimental estimate of the intrinsic strength of the noise.
“By implementing a motion reversal sequence with a Haar-random unitary transformation, the observed fidelity decay provides a direct experimental estimate of the intrinsic strength of the noise.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The noise superoperator Λ is independent of the specific target transformation but depends only on the duration of the experiment.
- May combine two claims. 4
- Classified as probably not a substantial claim. 6
May combine two claims.
Findingreasoning§ GENERALIZED FIDELITIES FOR CONTINUOUS-TIME WEAK NOISE · p. 6For small enough ε and long enough times t such that εt is fixed, the first term dominates and a specific expression can be written.
“Therefore for small enough ǫ and long enough times t such that ǫt is fixed the first term dominates and we can write”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingnone given§ APPENDIX: HAAR AVERAGED SUPEROPERATORS · p. 7Any superoperator Λ possesses exactly two linear U(D) invariants.
“Any superoperator ˆ Λ possesses exactly two linear U ( D ) invariants, i.e. the linear functionals on superoperator space which are invariant with respect to all transformation of the form ˆ Λ ↦→ ˆ U ˆ Λ ˆ U † :”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingreasoning§ APPENDIX: HAAR AVERAGED SUPEROPERATORS · p. 8From the fact that Tr(Λ_ave) = Tr(Λ), a specific expression is obtained.
“From the fact that Tr( ˆ Λ ave ) = Tr( ˆ Λ) we obtain”
Quote in the paperNot a substantial claimMay combine two claims.
Findingreasoning§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 2The Haar-averaged superoperator Λ_ave is U(D)-invariant and thus can be expressed as a depolarizing channel.
“As shown in the Appendix, the Haar-averaged superoperator ˆ Λ ave is U ( D ) -invariant and thus can be expressed as a depolarizing channel”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Premisecitation§ INTRODUCTION · p. 1Quantum process tomography (QPT) is the exact method for characterizing the noise affecting an implementation.
“The exact method for characterizing the noise affecting an implementation is quantum process tomography (QPT) [8].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [8] Nielsen M A, 2000, Quantum Computation and Quantum Information
Not checked in the free run.
Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Interpretationnone given§ Scalable Noise Estimation with Random Unitary Operators · p. 1By varying over different experimental arrangements and error-correction strategies, additional information about the noise can be determined.
“By varying over different experimental arrangements and error-correction strategies additional information about the noise can be determined.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S2 (checkable).
Interpretationreasoning§ Scalable Noise Estimation with Random Unitary Operators · p. 1The method should be immediately useful for evaluating the degree of control achievable in any prototype quantum processing device.
“our method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone given§ GENERALIZED FIDELITIES FOR CONTINUOUS-TIME WEAK NOISE · p. 6The ergodic hypothesis conditions are satisfied if the time-dependent dynamics can be modelled by a random walk on U(D) or by a trajectory on U(D) given by a deterministic dynamics with strong enough ergodic properties.
“These conditions are satisfied , for instance if the time-dependent dynamics t ↦→ U ( t ) can be modelled by a random walk on the group U ( D ) or by a trajectory on U ( D ) given by a certain deterministic dynamics with strong enough ergodic properties.”
Quote in the paperNot a substantial claimMay combine two claims.
Premisecitation§ EFFICIENT ESTIMATION OF THE AVERAGE GATE FIDELITY · p. 2The average fidelity depends only on the noise operator and can be expressed in a known form.
“Due to the invariance of the Fubini-Study measure the average fidelity depends only on the noise operator and can been expressed in the form [9, 10, 11]”
Quote in the paperSubstantial claim0 of 3 sources checkedMay combine two claims.
Interpretationcitation§ GENERALIZED FIDELITIES IN A DISCRETE-TIME SCENARIO · p. 4Due to the concentration of measure, for large D the fidelity loss under iterated motion reversal of a single sequence of random unitary operators will be exponentially close to the Haar-average, and hence the noise strength can be estimated with only a few experimental runs.
“Most importantly, due to the concentration of measure (14), for large D the fidelity loss under iterated motion reversal of a single sequence of random unitary operators will be exponentially close to the Haar-average, and hence the noise strength can be estimated with only a few experimental runs.”
Quote in the paperSubstantial claim
The authors state this as an assumption.
The cumulative noise for each U_j, or each step, has the same strength p_j = p.
“we can further simplify this result by assuming that the cumulative noise for each U j has the same strength p j = p”
“As before, we can simplify this result by assuming that the cumulative noise for each step has the same strength ( p j = p )”
The authors state this as an assumption.
The ergodic mean exists and is equal to the Haar average.
“We assume now the following ergodic hypothesis: a) the ergodic mean exists and is equal to the Haar average”
The authors state this as an assumption.
The fluctuations δL(t) around the ergodic mean are normal for long t.
“b) the fluctuations δ ˆ L ( t ) ≡ ˆ L ( t ) -ˆ L ave around ergodic mean are normal, i.e. for long t”
The authors state this as an assumption.
The noise superoperator Λ is independent of the specific target transformation but depends only on the duration of the experiment.
“We have throughout our analysis made the usual assumption that the noise superoperator Λ is independent of the specific target transformation but depends only on the duration of the experiment.”
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 uniform probability distribution over Clif_n comprises a unitary 2-design.
“One extremely useful property of Clif n , especially for noise estimation, is that the uniform probability distribution over Clif n comprises a unitary 2-design [17].”
From Characterizing quantum gates via randomized benchmarking (2012) · cites this paper as [17]
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Limitations of Linear Cross-Entropy as a Measure for Quantum Advantage
Gao, X., Kalinowski, M., Chou, C.-N., Lukin, M. D., Barak, B., & Choi, S. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.010334 · Cited by 63 · see citing works
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 50 · see citing works
Hashagen, A. K., Flammia, S. T., Gross, D., & Wallman, J. J. (2018) · Quantum · doi:10.22331/q-2018-08-22-85 · Cited by 56 · see citing works
Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates
Haferkamp, J., Montealegre‐Mora, F., Heinrich, M., Eisert, J., Gross, D., & Roth, I. (2022) · Communications in Mathematical Physics · doi:10.1007/s00220-022-04507-6 · Cited by 52 · see citing works
- Authors
- Emerson, J., Alicki, R., & Życzkowski, K.
- Affiliations
- Perimeter Institute (CA)
- University of Gdańsk (PL)
- Jagiellonian University (PL)
- Venue
- Journal of Optics B: Quantum and Semiclassical Optics
- Publisher
- IOP Publishing
- Published
- 2005-09-21
- Volume
- 7
- Issue
- 10
- Pages
- S347-S352
- ISSN
- 1464-4266 (ISSN-L)
- 1741-3575
- OpenAlex
- W2148118219
- Cited by
- 502
- References
- 24
- Citations by year
- 2026: 33 · 2025: 67 · 2024: 52 · 2023: 47 · 2022: 42 · 2021: 58 · 2020: 41 · 2019: 40 · 2018: 15 · 2017: 17 · 2016: 19 · 2015: 16 · 2014: 12 · 2013: 5 · 2012: 8
- FWCI
- 5.97
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Optics B Quantum and Semiclassical Optics (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- noise estimation, average gate fidelity, quantum process tomography, quantum control, non-unitary dynamics, superoperators, fidelity loss, error correction, scalable quantum devices
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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.
Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-10.
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