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Quantum advantages for Pauli channel estimation

Senrui Chen, Sisi Zhou, Alireza Seif, Liang Jiang

arXiv:2108.08488v2quant-ph

TL;DR

Pauli-channel eigenvalues are important for benchmarking and other quantum tasks, but their ultimate sample complexity has remained unresolved. This paper develops ancilla-assisted estimation and lower bounds across measurement models, showing an exponential advantage for entangled measurements and applying the protocol to benchmarking under reasonable noise assumptions.

  • Problem

    The ultimate sample complexity for estimating Pauli-channel eigenvalues remains incompletely characterized, despite their practical relevance to quantum benchmarking and fault-tolerant thresholds.

    Source: pp. 1–2

  • Method

    The paper develops an estimation protocol that divides the channel into ancilla-assisted Bell measurements and stabilizer-state syndrome measurements, and analyzes its sample complexity alongside lower bounds for restricted measurement models.

    Source: pp. 1, 3

  • Results

    O(n/ε^2) samples suffice with an n-qubit ancilla, whereas ancilla-free protocols require at least Ω(2^(n/3)) measurement rounds even with adaptive control and channel concatenation.

    Source: p. 1

  • Takeaways & Limitations

    The protocol also yields a benchmarking method for Pauli gates that is SPAM-error robust and exponentially more sample-efficient than ancilla-free schemes under reasonable noise assumptions.

    Source: p. 1

    The benchmarking method’s robustness and sample-efficiency claims are stated under reasonable noise assumptions.

    Source: p. 1

Abstract

from arXiv · show

We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an $n$-qubit Pauli channel to $\pm\varepsilon$ precision. We give an estimation protocol with an $n$-qubit ancilla that succeeds with high probability using only $O(n/\varepsilon^{2})$ copies of the Pauli channel, while prove that any ancilla-free protocol (possibly with adaptive control and channel concatenation) would need at least $\Omega(2^{n/3})$ rounds of measurement. We further study the advantages provided by a small number of ancillas. For the case that a $k$-qubit ancilla ($k\le n$) is available, we obtain a sample complexity lower bound of $\Omega(2^{(n-k)/3})$ for any non-concatenating protocol, and a stronger lower bound of $\Omega(n2^{n-k})$ for any non-adaptive, non-concatenating protocol, which is shown to be tight. We also show how to apply the ancilla-assisted estimation protocol to a practical quantum benchmarking task in a noise-resilient and sample-efficient manner, given reasonable noise assumptions. Our results provide a practically-interesting example for quantum advantages in learning and also bring new insight for quantum benchmarking.

Appendix A: Details about the proof of Theorem 1

The derivation uses Bell-state measurements on the ancilla subsystem and stabilizer-state syndrome measurements on the remaining qubits. Stabilizer labels are treated modulo the stabilizer group, and the resulting outcome probabilities encode the Pauli eigenvalues through a Walsh-Hadamard transform.

Appendix A: Details about the proof of Theorem 1

  • The input and measurement basis combine Bell states on the ancilla subsystem with stabilizer states and syndrome outcomes on the remaining subsystem.The joint input is |Ψ0⟩AB ⊗ |φS 0⟩C, and outcomes are indexed by Bell label v and syndrome e.Source: Appendix A · p. 7
  • Stabilizer states indexed by e and e + b represent the same state whenever b belongs to S, so labels are taken modulo S.The symplectic inner product uses an arbitrary representative of the corresponding coset.Source: Appendix A · p. 7
  • The joint measurement outcome distribution is Eq. (8), and its probabilities p(v, e) relate to λu⊕s through a Walsh-Hadamard transform.Source: Appendix A · p. 7

1. A tight lower bound for non-adaptive and non-concatenating strategies

This section establishes a tight sample-complexity lower bound for non-adaptive, non-concatenating estimation with k ancilla qubits. The proof reduces estimation to identifying a randomly chosen channel and bounds the information each measurement can provide.

1. A tight lower bound for non-adaptive and non-concatenating strategies

  • The lower-bound setting prepares an n + k qubit state for each channel sample and measures its joint output, without adapting to earlier outcomes or concatenating channel samples.These protocols include ancilla-free and n-qubit-ancilla strategies as special cases.Source: §1 · p. 7
  • Ω(n2^(n-k)) samples are required for non-adaptive, non-concatenating protocols using k ancilla qubits.The bound applies to arbitrary unknown n-qubit Pauli channels and is shown to be tight in the paper’s constructed protocol.Source: §1 · p. 7
  • The proof constructs channels indexed by a Pauli label and sign, then uses successful estimation to identify the randomly selected pair with high probability.Fano’s inequality yields an information requirement of Ω(n) about that pair.Source: §1 · p. 8
  • Each measurement round contributes at most O(2^(k-n)) mutual information about the channel index, yielding the sample lower bound.The argument reduces to pure-state inputs and rank-1 POVMs without increasing the available mutual information.Source: §1 · p. 8

2. A lower bound for adaptive but non-concatenating strategies

For adaptive but non-concatenating protocols with k ancilla qubits, the section proves a lower bound by reducing estimation to distinguishing two channel hypotheses. Its bound is weaker than the non-adaptive, non-concatenating bound established in the preceding section.

2. A lower bound for adaptive but non-concatenating strategies

  • Ω(2^((n-k)/3)) samples are required for adaptive, non-concatenating protocols with k ancilla qubits.The lower bound is described as possibly loose.Source: §2 · p. 10
  • The two hypotheses are a completely depolarizing channel and a channel Λa chosen uniformly from a family indexed by Pauli labels.An estimator meeting the theorem’s accuracy requirement would distinguish these hypotheses with high probability.Source: §2 · p. 10
  • Adaptivity allows each round’s input state and measurement to depend on previous outcomes, while each round still uses a separate channel sample.Source: §2 · p. 10
  • The proof compares outcome distributions for two equal-probability hypotheses and requires their total variation distance to be large for successful distinction.The bound TV(p1, p2) = O(N2^((k-n)/3)) gives the sample requirement.Source: §2 · pp. 10–11

3. A lower bound for the most general ancilla-free strategies

This section extends the ancilla-free lower bound to adaptive protocols that concatenate multiple channel uses before measurement. It models such strategies explicitly and proves that even this more general setting requires exponentially many measurement rounds.

3. A lower bound for the most general ancilla-free strategies

  • Concatenation models randomized-benchmarking strategies that apply varying numbers of noise channels before measuring decay values.The paper asks whether this practice also improves sample complexity; its bound says it does not remove the exponential requirement for ancilla-free estimation.Source: §3 · pp. 12–13
  • The protocol model permits adaptive choices of input states, measurements, and concatenation lengths, with classical processing of all outcomes to estimate the channel.N counts measurement rounds, not necessarily the number of channel samples consumed.Source: §3 · p. 13
  • Ω(2^(n/3)) measurement rounds are required by adaptive, concatenating ancilla-free protocols for Pauli eigenvalue estimation.The number of rounds is a lower bound on sample complexity when multiple channel copies may be concatenated in one measurement.Source: §3 · p. 13
  • The proof reduces estimation to distinguishing the completely depolarizing channel from a uniformly random channel in a Pauli-indexed family.Their outcome distributions have total variation distance O(N2^(-n/3)), forcing N = Ω(2^(n/3)).Source: §3 · p. 13

4. A lower bound for the most general entangled strategies

Even fully entangled strategies with unlimited quantum memory require at least Ω(n) samples to estimate the Pauli eigenvalues. The proof converts channel uses into measurements on Choi states and bounds the information those measurements can reveal.

4. A lower bound for the most general entangled strategies

  • Fully entangled protocols need at least Ω(n) samples to estimate the Pauli eigenvalues of an arbitrary n-qubit Pauli channel with high probability.The bound applies even with unlimited quantum memory and a final joint measurement.Source: §4 · p. 16
  • Teleportation stretching simulates each channel use using its Choi state, reducing an N-use protocol to a joint POVM on N Choi states.The simulation uses a Bell measurement and a conditional Pauli correction.Source: §4 · p. 16
  • A communication-based proof encodes a choice of (a, s) in a Pauli channel, so successful estimation lets the receiver identify that choice.The argument lower-bounds the required information using Fano’s inequality and upper-bounds it using Holevo’s theorem.Source: §4 · pp. 16, 18
  • The Choi state is maximally mixed on a 2^(2n−1)-dimensional space, giving entropy 2n−1 and yielding the Ω(n) sample lower bound.This entropy bound is used in the information argument.Source: §4 · p. 18

Appendix C: SPAM-robust ancilla-assisted Pauli gate benchmarking protocols

The appendix adapts ancilla-assisted Pauli-channel estimation to gate benchmarking with SPAM errors. Under stated noise assumptions, the protocol is SPAM-resilient and has O(n) sample complexity when the noise is weak; a rigorous complexity analysis is left for future work.

Appendix C: SPAM-robust ancilla-assisted Pauli gate benchmarking protocols

  • The guarantee assumes gate-independent, time-stationary, Markovian noise and ancilla noise confined to state preparation and measurement, independent of concatenation length.The appendix identifies negligible crosstalk during main-system gates and a comparatively long ancilla coherence time as relevant conditions.Source: Appendix C · pp. 18–20
  • The protocol estimates eigenvalues of the Pauli twirl of the gate noise using concatenated random Pauli gates, Bell-state preparation and measurement, and single-exponential fitting.The fitted quantity is A_aλ_a^m, with A_a independent of concatenation length m.Source: Appendix C · pp. 18–19
  • The appendix focuses on n ancillary qubits and leaves rigorous sample-complexity analysis and optimization of concatenation lengths and repetitions for future work.It notes a hybrid protocol for 0 < k < n ancillas but omits its details.Source: Appendix C · p. 19
  • Theorem 10 gives convergence to the true Pauli eigenvalues in a SPAM-resilient manner when concatenation lengths and circuit samples are sufficiently large.A_a is a noise-dependent constant independent of m; in the noiseless case A_a = 1.Source: Appendix C · pp. 19–20
  • Under weak noise, Algorithm 2 estimates all Pauli eigenvalues with O(n) samples, exponentially more sample-efficient than the ancilla-free protocol.The stated weak-noise condition is that both A_a and λ_a are bounded below by a constant.Source: Appendix C · p. 19

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