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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.
Overview
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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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21 No basis given · 17 Own interpretation · 8 Not checked yet · 1 Partly supported · 1 Own result · accuracy not yet measured
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“The more general fidelities are characterized by a universal exponential rate of fidelity loss.”
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“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.”
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“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”
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- The authors state this as an assumption: The cumulative noise for each U_j, or each step, has the same strength p_j = p.
In the paper
“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].”
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- Not checked yetcites [16] source
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“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”
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“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”
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“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.”
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“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.”
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“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”
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“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.”
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- 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.
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“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 ) .”
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“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”
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- 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.
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“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”
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“We remark that this result holds for general (possibly non-unital) noise.”
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“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.”
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“In the simplest case our method enables direct estimation of the average gate fidelity.”
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“While the scalability of our stochastic protocol makes it most relevant in large Hilbert spaces (when quantum process tomography is infeasible)”
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“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.”
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“Second, the number of experiments that must be carried out grows exponentially in the number of qubits D 4 = 2 4 n q .”
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“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].”
What the sources say
- Not checked yetcites [14] source
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“This universal form will allow us to evaluate the average fidelity for more generalized motion reversal protocols.”
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“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”
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“We show below that the overall noise strength and the associated accuracy of an implementation may be estimated by a scalable experimental method.”
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“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.”
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“Hence the gate fidelity can be determined directly from a standard readout (projective measurement) of the final state in the computational basis.”
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“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].”
What the sources say
- Not checked yetcites [12] Poźniak M, 1998, J. Phys. A: Math. Gen. source
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“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.”
What the sources say
- Not checked yetcites [12] Poźniak M, 1998, J. Phys. A: Math. Gen. source
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“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.”
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“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,”
What the sources say
- Not checked yetcites [13] Blume-Kohout R Emerson J, 2005
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“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”
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“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.”
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“We describe our system by the Markovian Master Equation [18, 19]”
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In the paper
“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”
What the sources say
- Not checked yetcites [2] source
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“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.”
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“Third, conventional numerical analysis of the tomographic data requires the manipulation of matrices of exponentially increasing dimension ( D 2 × D 2 ).”
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“We have described how generalized Haar-averaged fidelities may be directly estimated with only a few experimental measurements.”
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- 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.
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In the paper
“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.”
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“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.”
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- 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.
- It may state two separate claims as one. 4
- Probably not a substantial claim. 6
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In the paper
“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]”
What the sources say
- Not checked yetcites [9] source
- Partly supportedcites [10] sourceExperimental — accuracy not yet measured
The discussion on pp. 2–3 defines average fidelity by integrating over pure input states and derives a simplified way to calculate it for a single qubit. It does not state that the result depends only on a noise operator or attribute that simplification to invariance of the Fubini–Study measure; it describes fidelity between a unitary or anti-unitary map U and a general map M. (LLM, conf=0.83)
- Supportedcites [11] source
On p. 1, the paper defines the average using the uniform Haar measure and states that F(E,U) = F(U† ◦ E), so fidelity relative to the target gate is determined by the effective noise operation. On p. 2, it presents an explicit formula for that average gate fidelity. The paper does not use the term “Fubini–Study measure,” but the attributed point is supported. (LLM, conf=0.88)
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In the paper
“Therefore for small enough ǫ and long enough times t such that ǫt is fixed the first term dominates and we can write”
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In the paper
“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 † :”
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“From the fact that Tr( ˆ Λ ave ) = Tr( ˆ Λ) we obtain”
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“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”
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In the paper
“The exact method for characterizing the noise affecting an implementation is quantum process tomography (QPT) [8].”
What the sources say
- Not checked yetcites [8] Nielsen M A, 2000, Quantum Computation and Quantum Information
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In the paper
“By varying over different experimental arrangements and error-correction strategies additional information about the noise can be determined.”
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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.
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In the paper
“our method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device.”
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- Checkable: fails — It may be too vague to check against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may be too vague to check against evidence.
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
“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.”
plainconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The authors state this as an assumption.
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?
In the citing paper
“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]
Citations
Cites
Quantum Computation and Quantum Information
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895
Quantum computation and quantum information
Law, J. (2001) · ACM SIGSOFT Software Engineering Notes · doi:10.1145/505482.505499 · Cited by 20,450
On the generators of quantum dynamical semigroups
Lindblad, G. (1976) · Communications in Mathematical Physics · doi:10.1007/bf01608499 · Cited by 7,548
Completely positive dynamical semigroups of N -level systems
Gorini, V., Kossakowski, A., & Sudarshan, E. C. G. (1976) · Journal of Mathematical Physics · doi:10.1063/1.522979 · Cited by 3,998
Dynamical suppression of decoherence in two-state quantum systems
Viola, L., & Lloyd, S. (1998) · Physical Review A · doi:10.1103/physreva.58.2733 · Cited by 1,617
General teleportation channel, singlet fraction, and quasidistillation
Horodecki, M., Horodecki, P., & Horodecki, R. (1999) · Physical Review A · doi:10.1103/physreva.60.1888 · Cited by 965
A simple formula for the average gate fidelity of a quantum dynamical operation
Nielsen, M. A. (2002) · Physics Letters A · doi:10.1016/s0375-9601(02)01272-0 · Cited by 725
Knill, E., Laflamme, R., & Zurek, W. H. (1998) · Science · doi:10.1126/science.279.5349.342 · Cited by 663
Fault-tolerant quantum computation with constant error
Aharonov, D., & Ben-Or, M. (1997) · Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97 · doi:10.1145/258533.258579 · Cited by 631
Environment-Independent Decoherence Rate in Classically Chaotic Systems
Jalabert, R. A., & Pastawski, H. M. (2001) · Physical Review Letters · doi:10.1103/physrevlett.86.2490 · Cited by 571
Research works printed but not linked: 1
- Alicki R, 1987, Quantum Dynamical Semigroups and their Applications
Cited by
Sorted by citations
Trapped-ion quantum computing: Progress and challenges
Bruzewicz, C. D., Chiaverini, J., McConnell, R., & Sage, J. M. (2019) · Applied Physics Reviews · doi:10.1063/1.5088164 · Cited by 1,483
Characterizing quantum supremacy in near-term devices
Boixo, S., Isakov, S. V., Smelyanskiy, V. N., Babbush, R., Ding, N., Jiang, Z., Bremner, M. J., Martinis, J. M., & Neven, H. (2018) · Nature Physics · doi:10.1038/s41567-018-0124-x · Cited by 1,175
Suppressing quantum errors by scaling a surface code logical qubit
Google Quantum, A. I., Acharya, R., Aleiner, I., Allen, R., Andersen, T. I., Ansmann, M., Arute, F., Arya, K., Asfaw, A., Atalaya, J., et al. (2023) · Nature · doi:10.1038/s41586-022-05434-1 · Cited by 1,103
“Single qubit gates are characterized using parallel randomized benchmarking[31], while CZ gates are characterized with parallel cross-entropy benchmarking (XEB)[32].”
Randomized benchmarking of quantum gates
Knill, E., Leibfried, D., Reichle, R., Britton, J., Blakestad, R. B., Jost, J. D., Langer, C., Ozeri, R., Seidelin, S., & Wineland, D. J. (2008) · Physical Review A · doi:10.1103/physreva.77.012307 · Cited by 1,019
Scalable and Robust Randomized Benchmarking of Quantum Processes
Magesan, E., Gambetta, J., & Emerson, J. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.180504 · Cited by 797
Self-citation
Metadata
- 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 › 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 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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-10, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-10 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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