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Variational-State Quantum Metrology

Bálint Koczor, Suguru Endo, Tyson Jones, Yuichiro Matsuzaki, Simon C. Benjamin

arXiv:1908.08904v3quant-ph

TL;DR

Quantum metrology seeks more precise measurements with entangled probes, but environmental noise limits their performance. This paper variationally searches for noise-robust probe states, finding that nonsymmetric states can outperform symmetric ones in small systems. The authors also outline a possible near-term experimental implementation.

  • Problem

    Entangled probes can improve metrology, but noise degrades their sensitivity, and passive probes do not achieve superior fundamental scaling without active correction.

    Source: §1 · p. 3

  • Method

    The authors optimize parametrized probe states and sensing time for precision under a device-specific noise process using variational techniques.

    Source: §1 · p. 3, §3 · p. 7

  • Results

    Across simulations of up to 9 qubits, relaxing permutation symmetry yields near-optimal states that significantly outperform previously studied symmetric states under noise.

    Source: §1 · pp. 3–4, §7 · p. 21

  • Takeaways & Limitations

    The approach targets practically relevant small-qubit probes and can be tailored to hardware without specifying its noise model or using process tomography.

    Source: §1 · p. 4

    Verifying global optimality of the symmetry-breaking states is beyond the scope of this work, although simulations approximately found optimal states for non-symmetry-breaking noise channels.

    Source: §7 · p. 22

Abstract

from arXiv · show

Quantum technologies exploit entanglement to enhance various tasks beyond their classical limits including computation, communication and measurements. Quantum metrology aims to increase the precision of a measured quantity that is estimated in the presence of statistical errors using entangled quantum states. We present a novel approach for finding (near) optimal states for metrology in the presence of noise, using variational techniques as a tool for efficiently searching the classically intractable high-dimensional space of quantum states. We comprehensively explore systems consisting of up to 9 qubits and find new highly entangled states that are not symmetric under permutations and non-trivially outperform previously known states up to a constant factor 2. We consider a range of environmental noise models; while passive quantum states cannot achieve a fundamentally superior scaling (as established by prior asymptotic results) we do observe a significant absolute quantum advantage. We finally outline a possible experimental setup for variational quantum metrology which can be implemented in near-term hardware.

1. Introduction

Entangled probes can improve quantum-metrology precision, but environmental noise degrades their sensitivity. The paper uses variational optimization to seek noise-robust probes and reports gains from relaxing permutation symmetry.

1. Introduction

  • Entangled probes can beat the standard quantum limit, but under uncorrelated Markovian dephasing GHZ states achieve only standard-quantum-limit scaling.Prior results establish that passive noisy probes do not improve fundamental scaling unless actively corrected, although optimized probes can provide a constant-factor improvement.Source: §1 · p. 3
  • Variational optimization searches parametrized probe states using a cost function that quantifies precision in estimating the external field.The proposed procedure prepares a probe, exposes it to field and environmental noise, measures the output, and updates circuit parameters.Source: §1 · p. 3
  • Relaxing permutation symmetry yields highly entangled probe states that significantly outperform symmetric states under noise, despite the noise model being permutation symmetric.The authors derive a model in which these states passively correct first-order amplitude-damping events.Source: §1 · pp. 3–4
  • The approach targets practically relevant near-optimal states for small qubit systems, rather than asymptotically optimal states as N grows.The numerical exploration covers systems of up to 9 qubits.Source: §1 · pp. 3–4
  • The algorithm is designed to adapt to a device's noise without requiring a prespecified noise model or quantum process tomography.The authors also identify active error correction, pulse control, and multiparameter estimation as possible extensions.Source: §1 · p. 4

2. Precision in quantum metrology

This section describes how repeated measurements estimate a field parameter and how precision is quantified. It introduces classical and quantum Fisher information as tools for evaluating probe states and measurements.

2. Precision in quantum metrology

  • For a qubit probe sensing a magnetic field, the field strength is encoded as a relative phase through evolution generated by the collective angular momentum component Jz.Without noise, the probe undergoes a global rotation; with noise, the measured state is described by a density matrix.Source: §2 · pp. 4–5
  • Repeated projective measurements on identically prepared probes estimate the field parameter, with the inverse variance asymptotically approaching ν times the classical Fisher information.The estimation error is bounded by the Cramér-Rao bound.Source: §2 · p. 5
  • The classical Fisher information is calculated from measurement probabilities associated with the observable's eigenstates.The probabilities are obtained from the state density matrix and the projectors for measurement outcomes.Source: §2 · p. 5
  • Maximizing over generalized measurements for a fixed probe state gives the quantum Cramér-Rao bound.The quantum Fisher information is defined through the symmetric logarithmic derivative; measuring in its eigenbasis saturates the bound.Source: §2 · pp. 5–6
  • For mixed states, the symmetric logarithmic derivative can be constructed from the state's eigenvalues and eigenprojectors.The section also relates quantum Fisher information to the fidelity between states exposed to the same noise but different field strengths.Source: §2 · pp. 5–6

3. Variational state preparation for metrology

The proposed procedure prepares a parametrized probe, lets it evolve under the field and noise, and optimizes its parameters and sensing time for precision. Its evolution model accommodates continuous noise processes beyond those commuting with the field.

3. Variational state preparation for metrology

  • A shallow unitary encoder with single- and two-qubit gates prepares the parametrized probe, and its gate parameters are optimized externally.The authors propose an explicit encoder ansatz with a classically tractable number of parameters.Source: §3 · p. 6
  • The circuit diagram describes a procedure intended to find ψ(θ) for estimating the external field strength ω with best precision.The resulting state is exposed to the field and then analyzed to estimate ω.Source: §3 · p. 6, Figure 2 · p. 7
  • The evolution map models both the external field and non-unitary noise, with a continuous-time process that is not restricted to noise commuting with the field.The model uses the field superoperator and a completely positive trace-preserving generator for non-unitary dynamics; decay rates may also depend on time.Source: §3 · p. 7
  • The procedure varies encoder parameters and exposure time to maximize quantum Fisher information while balancing field sensitivity against noise.The circuit prepares a probe from the computational zero state, which then evolves in the environment before analysis.Source: §3 · pp. 6–7
  • The paper's experimental construction makes the analysis step explicit and does not rely on calculating quantum Fisher information.The authors defer implementation details to the experimental section.Source: §3 · p. 8

4. Numerical simulations

The simulations optimize probe parameters and sensing time under modeled noise, using quantum Fisher information or a fidelity-based approximation to evaluate precision. They assume ideal preparation and analysis in the main simulations, while also examining gate imperfections separately.

4. Numerical simulations

  • The main simulations assume perfect, negligible-duration encoder and analysis steps, with errors arising only during sensing; separate modeling shows the advantage remains attainable with imperfect preparation circuits.They assume γ tenc ≪ 1 and repeat experiments ν = T/t times.Source: §4 · p. 8
  • The simulations model the sensing evolution with Kraus maps and include dephasing, amplitude damping, and inhomogeneous Pauli errors.QuEST is used for exact numerical simulation of the device and noise processes.Source: §4 · p. 8
  • Quantum Fisher information evaluates the resulting state's metrological performance, avoiding explicit simulation of the analysis step.The simulations also estimate precision from fidelity between states evolved at two nearby field values, with δωt ≪ 1.Source: §4 · pp. 8–9
  • Finite differences can approximate the state derivative used in quantum Fisher information calculations, and may be more numerically stable and efficient than an eigendecomposition-based approximation.The authors compute the fidelity-based quantity by evaluating the circuit at two different evolutions.Source: §4 · p. 9
  • The simulations optimize the dimensionless precision over probe parameters and sensing time to obtain near-optimal noisy-metrology states.The dimensionless quantity is independent of the decay rate and total task time under the stated repetition assumption.Source: §4 · p. 9

5. Results

The results examine variationally optimized probe states for quantum metrology in systems of up to 9 qubits, including both symmetry-relaxed and permutation-symmetric states. They describe how probe structure and metrological performance depend on the noise model, with reported symmetric forms ranging from GHZ-like to squeezed-state-like.

5.1. Probe states

  • For N≤9, the proposed repeated-block ansatz has linearly many parameters and approximates optimal states across noise models, with additional blocks allowing better approximations.The reported B2B2B1B2B2B1 structure uses single-qubit X and Z rotations and nearest-neighbor controlled-Y rotations.Source: §5.1 · p. 10, Figure 3 · p. 11

5.2. Optimisation of the ansatz parameters

  • For N≤9, randomized adaptive coordinate descent searches without assumptions about the optimum; gradient descent and quantum natural gradient are proposed alternatives, with initialization near GHZ or squeezed states.The authors cannot guarantee global optimality in general, but compare against analytically known optima for some noise models and find good approximations.Source: §5.2 · p. 11

5.3. Probe states optimised against noise

  • Under amplitude damping, non-symmetric ansatz probes (N≥5) outperform every symmetric state by using longer sensing times, while all probes retain linear scaling.The optimized ansatz probes can spend significantly longer in the noisy environment than symmetric probes; the authors also report that their advantage persists under state-preparation gate imperfections.Source: Figure 4 · p. 13, §5.3 · p. 14
  • All dephasing probes retain linear scaling; optimized squeezed, symmetric, and ansatz states are nearly optimal, with improvements over GHZ or product states limited to a constant factor e≈2.72.The ansatz states differ negligibly from general symmetric states, supporting that the finite-depth circuit approximately finds the optimum; their sensing times lie between the GHZ and product-state optima.Source: §5.3 · pp. 12–13, Figure 4 · p. 13
  • With inhomogeneous Pauli errors, all probes show linear scaling, but ansatz states spend longer in the environment and outperform symmetric states for N≥6.The authors expect the ansatz states’ steeper slope to yield larger improvements as system size increases.Source: Figure 4 · p. 13, §5.3 · p. 15
  • For long-correlation-time Ornstein-Uhlenbeck noise, entangled probes improve scaling to N^c (1<c≤2), and GHZ states are optimal.The time-dependent noise buildup favors the shortest possible time in the environment.Source: Figure 5 · p. 15, §5.3 · p. 16

5.4. Analysis of the optimal states

  • Optimized general symmetric and ansatz states are highly entangled; breaking permutation symmetry improves sensitivity under amplitude damping, and also occurs for inhomogeneous Pauli errors.The nonsymmetric ansatz states are nearly GHZ-like under Pauli errors, whereas under amplitude damping the authors conjecture a near-optimal form for even N.Source: §5.4 · p. 17, Figure 6 · p. 18
  • For 8 qubits, a nonsymmetric state with coefficients c1≈0.77, c2≈0.55, and c3≈0.33 performs about 0.3% better than the optimized ansatz circuit.Its additional component |D⟩ resolves first-order effects of amplitude damping and improves performance over the symmetric Dicke-state counterpart.Source: §5.4 · p. 17
  • Under amplitude damping, the best symmetric state is a spin-up/down superposition with greater spin-down probability and a Wigner function similar to a GHZ state.Source: §5.4 · pp. 18–19, Figure 7 · p. 19
  • For permutation-symmetric 9-qubit states, the dephasing-optimized Wigner function resembles a squeezed state, while the Ornstein-Uhlenbeck case is very close to a GHZ state.Source: Figure 7 · p. 19
  • For inhomogeneous Pauli errors, GHZ states are nearly optimal, and the optimized state's Wigner function is similar to that of a GHZ state.Source: §5.4 · p. 19, Figure 7 · p. 19

6. Possible experimental implementation

The proposed setup variationally tunes encoder and decoder circuits to find a probe state and projective-measurement strategy for estimating ω. The device estimates precision from repeated measurements, while universal circuits would in principle allow the optimal state and measurement strategy.

6. Possible experimental implementation

  • Encoder and decoder parameters are varied to optimize the precision for estimating ω after the probe interacts with the field and noisy environment.The setup uses repeated device executions to estimate precision.Source: §6 · p. 19
  • The decoder maps evolution information into computational-basis measurement probabilities, equivalently measuring in a decoder-dependent effective basis.The effective observable is O(θd) := U†DJzUD.Source: §6 · p. 20
  • Repeated measurements estimate probabilities and their derivatives, allowing the device to estimate the Cramér–Rao precision targeted by the optimization.The derivatives are approximated by repeating experiments at ω + δω and taking a finite difference.Source: §6 · p. 20
  • If encoder and decoder circuits span SU(2^N), the setup can in principle combine an optimal sensing state with its corresponding best measurement strategy.The proposal identifies superconducting qubits as potentially suitable for an experimental demonstration.Source: §6 · pp. 20–21

7. Discussion and conclusion

The authors propose variational methods for finding noise-robust metrology states, and simulations across noise models and systems up to nine qubits find near-optimal, often nonsymmetric states that can outperform known symmetric states. They note that global optimality of symmetry-breaking states has not been verified and outline extensions to noisy preparation hardware and error correction.

7. Discussion and conclusion

  • Compared with Ref. [85], the approach addresses noise and is intended to extend beyond very small quantum-optics systems through variational techniques that can run on quantum hardware.The cited classical method is limited to average photon number below two because of computational complexity and does not account for noise.Source: §7 · p. 21
  • Across simulations of up to 9 qubits and several noise models, a fixed ansatz finds previously known optimal states and near-optimal families that outperform previously known symmetric states.Relaxing probe-state permutation symmetry gives significant improvements; symmetric states studied in detail in this context are cited as [39] [41] [42] [43].Source: §7 · p. 21
  • Global optimality of the symmetry-breaking states remains outside this work’s scope, although simulations approximately found optimal states for non-symmetry-breaking noise channels.Source: §7 · p. 22
  • The resulting optimal states are usually highly entangled, but not necessarily maximally entangled; the authors also outline a possible near-term hardware realization.The entanglement observation is noted as consistent with.Source: §7 · p. 22
  • Proposed extensions include noisy state-preparation hardware, with or without error mitigation, and combining the optimization with error-detection and error-correction concepts [28] [29] [31] [34] [36].Source: §7 · p. 22

Appendix A. Deriving the dimensionless precision

The appendix rescales sensing time and the field offset to express the precision in terms of a unit decay rate. The resulting dimensionless maximum precision is independent of the noise model’s decay rate.

Appendix A. Deriving the dimensionless precision

  • The derivation starts from precision expressed through the fidelity between states evolved at ω = 0 and ω = δω from the probe state.Source: Appendix A · p. 22
  • Rescaling time as t → t′/γ and field offset as δω → δ′ωγ gives dynamics equivalent to a unit decay rate γ → 1.Source: Appendix A · pp. 22–23
  • The maximum precision varies with decay rate as f(γ) = c/γ, leaving the constant factor c as the degree of freedom.Source: Appendix A · p. 23
  • The dimensionless precision γ/T(∆ω)^−2_max is independent of the decay rate of the noise model.Source: Appendix A · p. 23

Appendix B. Kraus operators of the Ornstein-Uhlenbeck noise

The appendix describes the Ornstein–Uhlenbeck Kraus map through a time-dependent probability and shows that it is equivalent to simple dephasing. Its dephasing probability is also time-dependent.

Appendix B. Kraus operators of the Ornstein-Uhlenbeck noise

  • The Kraus representation of Ornstein–Uhlenbeck noise is given in terms of single-qubit Kraus operators.Source: Appendix B · p. 23
  • The Ornstein–Uhlenbeck Kraus map is equivalent to simple dephasing with time-dependent probability p(t) = [1 − e^−f(t)]/2.Here f(t) := γ[t + (e^−λt − 1)/λ]/2, and q(t) = e^−f(t).Source: Appendix B · p. 23

Appendix C. Improved performance of non-symmetric states

Under amplitude damping, non-symmetric states outperform their symmetric analogues because first-order decay events add informative measurement outcomes. These outcomes expand the measurement basis and increase the classical Fisher information available to the non-symmetric scheme.

Appendix C. Improved performance of non-symmetric states

  • For even N, optimized states lie close to |ψa⟩, whose |D⟩ component breaks permutation symmetry and comprises two-excitation configurations.The symmetric Dicke components are |J, J⟩ and |J, −J⟩, defined for even N with J = N/2.Source: Appendix C · p. 23
  • For N = 6 and N = 8, the performance of |ψa⟩ almost exactly coincides with that of the variationally determined states.Source: Appendix C · p. 23
  • For N = 8, the optimized non-symmetric state trades higher contributions from the last 16 |Aj⟩ bases against lower contributions from the first two bases.The caption attributes this trade-off to increasing the |D⟩ component while decreasing the absolute values of c1 and c3.Source: Figure C1 · p. 25
  • First-order T1 relaxation gives nonzero probabilities to additional measurement bases, including |S⟩ for the symmetric case and 2N bases for the non-symmetric case.The non-symmetric bases correspond to individual-qubit flips; their probabilities arise only from first-order T1 relaxation.Source: Appendix C · pp. 24–25
  • The non-symmetric measurement is as sensitive as the symmetric one, while its N distinguishable |Aj⟩ bases yield classical Fisher information scaling proportional to N.The additional information comes from the part of the state affected by amplitude damping and can be comparable to the shared zeroth-order contribution in the non-symmetric scheme.Source: Appendix C · p. 26

Appendix D. Effect of non-ideal quantum circuits

The appendix tests how imperfect preparation circuits affect the performance of the state families. Despite deeper preparation circuits, ansatz states retain their superior performance under the stated gate noise.

Appendix D. Effect of non-ideal quantum circuits

  • The simulations assign depolarizing error probabilities of 10^-4 to single-qubit gates and 10^-3 to two-qubit gates.The authors describe these rates as comparable to currently existing technology [86].Source: Appendix D · p. 26
  • The deeper ansatz circuits experience a more substantial performance decrease from gate noise than GHZ-state preparation circuits.This decrease is reflected by comparing the solid and dashed lines in Figure D1.Source: Appendix D · p. 26, Figure D1 · p. 27
  • Ansatz states retain superior performance under imperfect preparation circuits, despite their deeper circuits.The figure compares noisy-circuit results with ideal unitary-preparation results shown as dashed lines.Source: Appendix D · p. 27, Figure D1 · p. 27

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