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A variational toolbox for quantum multi-parameter estimation

Johannes Jakob Meyer, Johannes Borregaard, Jens Eisert

arXiv:2006.06303v1quant-ph

TL;DR

Designing near-optimal probes and measurements for noisy multi-parameter estimation is challenging because the search over high-dimensional quantum states becomes classically infeasible in practice. The paper presents a variational approach that updates probe and measurement parameters and establishes parameter-shift rules for noisy evolutions. Its simulations report improved sensing procedures in noisy settings, including cases that outperform GHZ sensing and standard Ramsey spectroscopy.

  • Problem

    Finding suitable probes and measurement schemes for noisy metrology is difficult because optimizing over high-dimensional quantum states soon becomes classically infeasible in practice.

    Source: p. 1

  • Method

    The approach uses a variational quantum circuit to update probe-state and measurement parameters based on estimation quality, with analytic derivatives computed using parameter-shift rules for noisy evolutions.

    Source: p. 2

  • Results

    At increasing noise levels, the reported optimized sensing procedures outperform both GHZ sensing and standard Ramsey spectroscopy.

    Source: p. 4

  • Takeaways & Limitations

    The approach learns variational parameters for multi-parameter estimation schemes and can be tailored to the sensing problem and available hardware.

    Source: p. 4

    The approach does not cover all possible metrological tasks, and parameter-shift applicability depends on the exact encoding unitary.

    Source: pp. 3–4

Abstract

from arXiv · show

With an ever-expanding ecosystem of noisy and intermediate-scale quantum devices, exploring their possible applications is a rapidly growing field of quantum information science. In this work, we demonstrate that variational quantum algorithms feasible on such devices address a challenge central to the field of quantum metrology: The identification of near-optimal probes and measurement operators for noisy multi-parameter estimation problems. We first introduce a general framework which allows for sequential updates of variational parameters to improve probe states and measurements and is widely applicable to both discrete and continuous-variable settings. We then demonstrate the practical functioning of the approach through numerical simulations, showcasing how tailored probes and measurements improve over standard methods in the noisy regime. Along the way, we prove the validity of a general parameter-shift rule for noisy evolutions, expected to be of general interest in variational quantum algorithms. In our approach, we advocate the mindset of quantum-aided design, exploiting quantum technology to learn close to optimal, experimentally feasible quantum metrology protocols.

Appendix A: Analytic gradients for noisy quantum circuits

Appendix A establishes when parameter-shift rules can compute gradients for noisy quantum circuits and certain parameterized channels. It extends the rule from suitable unitary evolutions to convex combinations of channels, including cases useful for channel simulation.

Motivation

  • Variational algorithms use classical feedback to train parameterized circuits, and gradient-based optimization motivates the appendix’s analytic-gradient results.The text notes that unbiased gradient estimates suffice for convergence and describes stochastic extensions for more intricate evolutions.Source: Appendix A · p. 7

Convex combinations of channels

  • Dephasing and depolarizing channels are cited as examples within the convex-combination framework for parameter-shift rules.The appendix identifies these rules as potentially useful for error-channel simulations.Source: Appendix A · pp. 7–8

Unitary evolutions

  • For a unitary generated by an operator with two distinct eigenvalues, the gradient is a linear combination of evaluations at shifted parameter values.The shift scale r is half the eigenvalue difference; the text also notes parameter-shift parametrizations for Pauli-word and Gaussian elementary operations.Source: Appendix A · pp. 7–8

Noisy circuits and general prerequisites

  • Noisy unitaries followed by parameter-independent noise can be differentiated with the same parameter-shift rule as the corresponding perfect unitary.The rule also applies when such channels are composed with other channels independent of the shifted parameter.Source: Appendix A · pp. 7–8

Convex combinations of channels

  • Convex combinations of two quantum channels admit a gradient rule independent of the mixture parameter, using channel evaluations at two distinct parameter values.The proof uses two realizations of the channel and extends analogously to convex combinations of more than two channels.Source: Appendix A · pp. 8–9

Appendix B: Conditions for the parameter-shift rule

Appendix B states when derivatives can be evaluated directly on sensing platforms or by emulation. The parameter-shift rule requires control of the measured parameters and applies under specified encoding conditions.

Parameter control

  • Direct derivative evaluation requires control to shift the measured parameter, which is usually impossible on the sensing platform unless the shift can be added before encoding.Imperfect implementation of the shift itself adds noise to optimization, even though the approach accounts for noise in state preparation, encoding, and the POVM.Source: Appendix B · p. 10

Conditions for the parameter-shift rule

  • The parameter-shift rule applies to separate discrete-variable encodings generated by Hamiltonians with two distinct eigenvalues and to separate Gaussian phase shifts, squeezings, and displacements.For discrete variables, the parameters can be reparameterized and the CFIM transformed back to the original parameters.Source: Appendix B · p. 10

Evaluation cost

  • Either finite differences or parameter-shift methods require evaluating the setup at 2d+1 parameter-space points to calculate the CFIM.Stochastic methods can cover more intricate evolutions but require evaluating the setup at more points.Source: Appendix B · p. 10

Appendix C: Sensing emulation

Appendix C describes two ways to emulate error channels: use a larger system for a unitary dilation, or sample unitaries for mixed-unitary channels.

Error-channel emulation

  • Stinespring dilation realizes any error channel as a unitary evolution on a larger system, using ancillary qubits or modes.The text presents this as one route for emulating error channels.Source: Appendix C · p. 10
  • Mixed-unitary channels can be emulated by repeatedly sampling a unitary from the channel’s distribution and executing the circuit.Examples include bit-flip, depolarizing, and dephasing channels.Source: Appendix C · p. 10

Hardware dependence

  • More complicated error channels can be engineered with closer hardware access, but available capabilities depend strongly on the underlying technology.Source: Appendix C · p. 10

Appendix D: Extensions

Appendix D announces three extensions to the variational approach.

Extensions

  • The appendix introduces three extensions of the variational approach, without specifying them in this passage.Source: Appendix D · p. 10

1. Bayesian quantum metrology: Including prior knowledge

Prior knowledge about parameter distributions leads to a Van Trees bound that can be used to construct a Bayesian cost function. The approach uses expected Fisher information and can incorporate the prior through repeated sensing runs.

1. Bayesian quantum metrology: Including prior knowledge

  • Prior knowledge about the parameter distribution P(φ) leads to the more general Van Trees inequality.The Cramér-Rao bound depends on the underlying parameters φ.Source: §1 · p. 10
  • The Bayesian cost uses the expected CFIM and the prior's CFIM, but only the expected CFIM scales with sample number n.The prior contribution can be neglected unless estimation is sample-frugal.Source: §1 · p. 10
  • On a sensing platform, the variational approach automatically accounts for Bayesian parameters; in simulation, runs can draw fresh φ values from P(φ).The CFIM can then be estimated from the results of those runs.Source: §1 · p. 11

2. Including time dependence

When both the encoded parameters and noise depend on exposure time, the parameter-shift rule no longer applies. Finite differences or, in simulation, direct derivatives can supply the time-dependent derivatives used to obtain the CFIM and cost.

2. Including time dependence

  • When noise and parameter encoding both depend on exposure time t, the parameter-shift rule is unavailable, but finite differences can obtain derivatives on-platform or in emulation.Simulation also permits direct derivative calculation.Source: §2 · p. 11
  • The time derivative of the probabilities has channel-dependent and state-dependent contributions; the state-dependent part can reuse already calculated output probabilities.The channel-dependent part requires the time derivative of the error channel on the classical optimization side.Source: §2 · p. 11
  • If parameter encoding is parameter-shift differentiable, no extra time derivatives of partial derivatives are needed; otherwise those derivatives must also be calculated.The additional terms account for error dependence and encoded-state dependence.Source: §2 · p. 11
  • The reformulated derivatives are expectation values that can be used with Eq. (3) to obtain the CFIM and then the cost function.The expressions are more complicated.Source: §2 · p. 12

3. Evaluation of the quantum Cram´er-Rao bound

The quantum Cramér-Rao bound provides a precision limit independent of the chosen POVM, but evaluating it for arbitrary mixed probes is difficult. Pure probes permit a simpler QFIM calculation that can be measured and optimized, though current devices generally require classical simulation for that optimization.

3. Evaluation of the quantum Cramér-Rao bound

  • The quantum Cramér-Rao bound is a more fundamental precision bound independent of the chosen POVM.It uses the quantum Fisher information matrix (QFIM).Source: §3 · p. 12

3. Evaluation of the quantum Cram´er-Rao bound

  • For arbitrary mixed probe states, calculating the QFIM requires full tomography or solving an infeasibly large semidefinite program.The QFIM gives the fundamental precision limit for the sensing problem.Source: §3 · p. 12
  • For pure probes, the QFIM reduces to the generators' fourfold covariance matrix and can be estimated from measurements of observables H_j and their products.The resulting quantity can also be differentiated via the parameter-shift rule and used in an optimization cost.Source: §3 · p. 12
  • Current quantum devices usually lack the purity required for this QFIM optimization, so it must be done by classical simulation; pure states benchmark performance when noise is suppressed.Convexity ensures maximum Fisher information is attained on pure states.Source: §3 · p. 12

1. Ramsay spectroscopy

The Ramsay spectroscopy simulations use a three-qubit preparation circuit and a locally unitary-parameterized measurement. The optimization varies dephasing and uses repeated gradient-descent runs, selecting the best outcome.

1. Ramsay spectroscopy

  • The preparation circuit parametrizes any pure three-qubit state with the minimal number of parameters.The encoding includes dephasing noise D with probability p.Source: §1 · p. 13
  • The measurement is parametrized by a local unitary transformation, with R_P(θ) denoting a rotation generated by Pauli word P.The preparation and measurement circuits were used for the numerical simulations.Source: §1 · pp. 12–13
  • Optimization fixed encoded phases at π/6, varied dephasing probability p from 0 to 0.5, and selected the best result across repeated 1,000-step gradient-descent runs.Weights were randomly initialized over [0, 2π], the initial step size was 0.05, and 1 × 10^-10 × I regularized the CFIM before inversion.Source: §1 · p. 13

2. NV trilateration

The section formulates three-NV spin-position estimation under dipolar coupling and compares local and shallow-entangled probe preparations with noisy gates. It also specifies the noise model and optimization procedure used in the simulations.

2. NV trilateration

  • Three NV centers at characterized positions estimate an unknown target-spin position using the NV centers’ ms = 0 and ms = −1 ground states.Source: §2 · p. 13
  • The model uses a secular dipole interaction, neglects NV–NV interactions assumed decouplable by pulse sequences, and simplifies the Hamiltonian with DEER.Source: §2 · p. 13
  • Sympy computes the phase-position Jacobian, which is inverted for the optimization; gradient descent runs for 1000 steps, with the best of multiple runs selected.Source: §2 · pp. 14–15
  • The comparison uses local preparation and shallow entangling preparation, which adds two entangling operations and can create a GHZ state.Source: §2 · p. 14
  • Both preparations use local unitaries to parametrize measurement, while equal single- and two-qubit depolarizing gate noise models the gate noise.Source: §2 · p. 14
  • The accumulated phase is fixed at π/2 and encoding dephasing at pe = 0.1, while gate depolarization pg varies from 0 to 0.1.Source: §2 · pp. 14–15

Appendix F: CO2 Emission Table

The appendix reports energy and CO2 figures for the simulations and transport, and whether emissions were offset.

Appendix F: CO2 Emission Table

  • 167.2 kWh is reported for total simulation energy consumption, based on 29,071 kernel hours at 5.75 W per kernel.Source: Appendix F · p. 15
  • Transport CO2 emissions are reported as 0 kg.Source: Appendix F · p. 15
  • 94 kg is the reported total CO2 emission, and the table marks emissions as offset.Source: Appendix F · p. 15

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