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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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Written from the paper’s full text.
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
21 No basis given · 17 Own interpretation · 9 Not checked yet · 1 Own result
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 more general fidelities are characterized by a universal exponential rate of fidelity loss.”
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
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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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
“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”
plaindirect- Asserted: passes — The authors put it forward as true.
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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].”
What the sources say
- Not checked yetcites [16] source
hedgeddirect- 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
“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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In the paper
“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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In the paper
“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.”
plain- 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 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.”
hedgedcorrelational- Asserted: passes — The authors put it forward as true.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“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”
hedgedcorrelational- 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
“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.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- The 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
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
“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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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Any 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 † :”
plain- 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.
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
“From the fact that Tr( ˆ Λ ave ) = Tr( ˆ Λ) we obtain”
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.
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
“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”
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
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
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial 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
“By varying over different experimental arrangements and error-correction strategies additional information about the noise can be determined.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
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
“our method should be immediately useful for evaluating the degree of control that is achievable in any prototype quantum processing device.”
plain- Asserted: passes — The authors put it forward as true.
- 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.
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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
“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
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.
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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
“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
Items 501–502 of 502
Santos, C. E. D. S., & de Freitas, R. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-39895-6_31
Optimal control and quantum simulations in superconducting quantum devices
Egger, D. J. (2014) · Publications of the UdS (Saarland University) · doi:10.22028/d291-23000
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
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