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Quantum natural gradient generalised to non-unitary circuits

Bálint Koczor, Simon C. Benjamin

arXiv:1912.08660v4quant-ph

TL;DR

The paper addresses how to optimize variational quantum circuits when noise and non-unitary operations produce arbitrary mixed or pure states. It generalizes quantum natural gradient through completely positive maps and uses quantum Fisher information as the state-space metric. Simulations indicate improved performance over simple gradient descent, while mixed-state imaginary-time methods can remain suboptimal and fail to reach the true optimum.

  • Problem

    Existing variational optimization methods were developed mainly for ideal unitary circuits or restricted state representations, limiting their direct application to noisy circuits and arbitrary density operators.

    Source: INTRODUCTION · pp. 1–2

  • Method

    The paper models circuits as completely positive maps and applies a Riemannian natural-gradient update using the quantum Fisher information matrix for mixed or pure states.

    Source: INTRODUCTION · p. 2, §III · p. 4

  • Results

    Numerical simulations show that the proposed approach significantly outperforms simple gradient descent, while mixed-state imaginary-time evolution can be suboptimal and fail to converge to the true optimum.

    Source: §VI · p. 11

  • Takeaways & Limitations

    Quantum Fisher information provides a unified metric for natural-gradient optimization across arbitrary quantum states and noisy or otherwise non-unitary circuits.

    Source: INTRODUCTION · p. 2

Abstract

from arXiv · show

Variational quantum circuits are promising tools whose efficacy depends on their optimisation method. For noise-free unitary circuits, the quantum generalisation of natural gradient descent was recently introduced. The method can be shown to be equivalent to imaginary time evolution, and is highly effective due to a metric tensor reconciling the classical parameter space to the device's Hilbert space. Here we generalise quantum natural gradient to consider arbitrary quantum states (both mixed and pure) via completely positive maps; thus our circuits can incorporate both imperfect unitary gates and fundamentally non-unitary operations such as measurements. Whereas the unitary variant relates to classical Fisher information, here we find that quantum Fisher information defines the core metric in the space of density operators. Numerical simulations indicate that our approach can outperform other variational techniques when circuit noise is present. We finally assess the practical feasibility of our implementation and argue that its scalability is only limited by the number and quality of imperfect gates and not by the number of qubits.

INTRODUCTION

Variational quantum circuits optimize parameterized quantum states, but noise complicates methods developed for ideal unitary circuits. The paper extends quantum natural-gradient ideas toward noisy and general quantum-state optimization.

INTRODUCTION

  • Variational algorithms use tractable parameters to search a small relevant subspace of an exponentially larger Hilbert space.Source: INTRODUCTION · p. 1
  • VQE minimizes an expectation-value cost such as energy using classical optimization of parameters estimated by a quantum processor.Source: INTRODUCTION · p. 1, §I · p. 3
  • Imaginary-time evolution and natural-gradient methods improve optimization by using a parameter-sensitivity matrix that accounts for the geometry of quantum states [34].Source: INTRODUCTION · p. 1
  • The paper proposes a variational optimization method for arbitrary mixed or pure states, including non-unitary transformations such as measurements.Source: INTRODUCTION · p. 2

I. OPTIMISING IDEALISED VARIATIONAL QUANTUM CIRCUITS

  • For ideal unitary circuits, the metric A corrects gradients for the non-uniform and co-dependent effects of parameters and is related to the Fubini–Study and classical Fisher metrics [34].Source: §I · p. 3
  • Imaginary-time evolution converges to the ground state when the initial state overlaps it, whereas noise-free formulas can become suboptimal on noisy circuits.Source: §I · p. 3

II. VARIATIONAL ALGORITHMS WITH NON-UNITARY CIRCUITS

The paper models realistic circuits as parameterized completely positive maps and develops a natural-gradient framework for their resulting density operators. Quantum Fisher information measures parameter sensitivity for these arbitrary states.

II. VARIATIONAL ALGORITHMS WITH NON-UNITARY CIRCUITS

  • Realistic variational circuits are represented as completely positive maps ρ(θ)=Φ(θ)ρ0, approximately decomposed into imperfect parameterized gates.Source: §II · p. 3
  • The framework assumes continuity in each parameter, covering Markovian decoherence, parameter-independent measurements, parameter-dependent decoherence, and continuous-variable systems.Source: §II · p. 3
  • Unlike mixed-state imaginary-time evolution, the proposed Riemannian gradient descent uses quantum Fisher information and is designed to converge to the true optimum.Source: §II · p. 4

III. QUANTUM FISHER INFORMATION AND NATURAL GRADIENT

  • Quantum Fisher information is introduced as a measure of how parameter changes affect general, potentially mixed quantum states in variational circuits.Source: §III · p. 4

A. Sensitivity to parameters via the quantum Fisher information

Quantum Fisher information captures how parameter changes alter measurement statistics and provides a geometry for arbitrary density-operator families. It connects the mixed-state method to classical Fisher and pure-state Fubini–Study geometries.

A. Sensitivity to parameters via the quantum Fisher information

B. Quantum Fisher information matrix

C. Geometry of quantum states and metric tensors

D. Natural gradient descent for arbitrary quantum states

The proposed update minimizes a variational objective by correcting its parameter gradient with the quantum Fisher information matrix. This gives a natural-gradient rule for arbitrary mixed or pure quantum states produced by continuous quantum mappings.

D. Natural gradient descent for arbitrary quantum states

E. Natural gradient for idealised unitary circuits

The quantum Fisher information natural-gradient method recovers established pure-state results while extending optimisation to more expressive non-unitary objectives. In noisy simulations, it approaches the optimum more accurately and precisely than gradient descent and mixed-state imaginary-time evolution.

E. Natural gradient for idealised unitary circuits

  • For pure states, the quantum Fisher information reduces to four times the Fubini–Study metric, making the natural-gradient update identical to simulated imaginary-time evolution.With step size λ = 4∆t, the general update recovers the pure-state evolution used in prior approaches.Source: E. Natural gradient for idealised… · p. 6
  • In arbitrary pure states, quantum Fisher information is more general than classical Fisher information, while the two coincide when the state encodes a classical probability distribution.Thus the classical-Fisher-information formulation is recovered only in the probability-distribution special case.Source: E. Natural gradient for idealised… · p. 6

F. Fundamental improvements

  • The method extends beyond earlier pure-state and observable objectives through continuous CPTP maps, which support mixed states and exponentially more Fourier components in the number of gates.The stated expansion is O(2^4ν) for comparable unitary circuits versus O(2^8ν) for CPTP-map circuits, with local CPTP maps implementable efficiently.Source: F. Fundamental improvements · p. 6
  • Its convergence guarantee covers general objectives with Lipschitz-continuous gradients, including nonlinear functions of expectation values, cross entropy, and log-likelihoods.These objectives extend beyond the linear observable-based functions targeted by previous approaches.Source: F. Fundamental improvements · p. 6
  • The metric regulates ill-behaved non-unitary parameter evolution, while natural-gradient dynamics are parametrisation-independent and avoid local minima in the stated example.The single-qubit analysis reports exponentially faster convergence than simple gradient descent.Source: F. Fundamental improvements · pp. 6–7

IV. NUMERICAL SIMULATIONS

  • The noisy benchmark uses a spin-chain Hamiltonian and a mixed-state ansatz with depolarising errors after every single- and two-qubit gate.Two-qubit errors are set ten times the single-qubit error probability, and the circuit contains continuous rotations and nearest-neighbour couplings.Source: §IV · p. 7
  • The quantum Fisher information method achieves the smallest mean energy distance and standard deviation among the compared noisy-circuit optimisers.After 30 steps, it is more precise and accurate than gradient descent and mixed-state imaginary-time evolution, whose estimate remains biased away from the optimum.Source: §IV · p. 7

V. EFFICIENT APPROXIMATION AND EXPERIMENTAL IMPLEMENTATION

The paper develops an efficient approximation to the quantum Fisher information for noisy variational circuits and evaluates its accuracy, resource demands, and scalability. The approximation remains accurate in relevant noise regimes while shifting practical limits toward gate count and gate quality rather than qubit number.

Efficient approximation

  • The approximation error is O(1−fid) and decreases as O(2^-N) with qubit number under the stated near-term noise assumptions.For fixed fidelity, errors decrease beyond N = 2; the derivation assumes independent gate errors and approximately parameter-independent error probabilities.Source: §V · pp. 8–9

Experimental implementation

  • The approximation uses Hilbert-Schmidt scalar products of density-operator derivatives, which can be estimated experimentally with SWAP tests while omitting an approximately constant inverse-fidelity step-size factor.Result 2 targets mixed states produced by independent gate errors on near-term hardware.Source: §V · p. 8

Accuracy

  • Errors below 10^-3 occur for fid > 0.9, and the approximation closely tracks the exact quantum-Fisher-information evolution.The comparison in Fig. 4(b) shows the approximated evolution closely following the exact natural-gradient trajectory, unlike noisy pure-state imaginary-time evolution.Source: §V · p. 9, Figure 4 · p. 9

Scalability

  • For constant measurement overhead Ns = 16, the maximum qubit count is approximately 230 at perr = 10^-3 and 2310 at perr = 10^-4 for circuits with Ng = aN and a ≈ 3.These bounds follow from the exponential fidelity decay with imperfect-gate count.Source: §V · p. 9

Limitations

  • The method’s practical limits arise from exponential fidelity decay with imperfect-gate count, which increases measurement overhead and affects gradient and cost-function estimation as well.The required ancilla-measurement overhead scales as Ns = O(fid^-4), while gradient and cost-function estimation also incur fidelity-dependent overhead.Source: §V · pp. 9–10

Numerical implementation

  • Replacing exact FQ with the efficient approximation preserves optimisation performance while reducing computational cost and enabling simulations with more qubits.The 60-step simulation in Fig. 5 shows no significant performance degradation, and the approximation is computationally preferable.Source: §V · p. 10, Figure 5 · p. 10

Comparison

  • Compared with Hadamard-test-based imaginary-time estimation, Result 2 has stronger error suppression with qubit number and provides a more error-robust approximation at constant resource overhead.Hadamard tests use fewer qubits, but their approximation error does not decrease with qubit number in the same way.Source: Figure 4 · p. 9, §V · p. 10

VI. DISCUSSION AND CONCLUSION

The paper extends quantum natural-gradient optimisation from noise-free unitary circuits to arbitrary mixed and pure states represented as density matrices. Simulations support improved optimisation under noise, while the proposed approximation remains practical when imperfections are limited by gate count and quality.

Contribution

  • Quantum Fisher information corrects the gradient for the non-uniform effects of parameters on arbitrary quantum states represented as density matrices.This extends the quantum natural-gradient framework beyond the general unitary, noise-free setting.Source: §VI · p. 10

Results

  • Noisy-circuit simulations show the approach significantly outperforms simple gradient descent, while mixed-state imaginary-time evolution can remain suboptimal and fail to reach the true optimum.The paper therefore presents the generalisation as a robust candidate for noisy or otherwise non-unitary circuits.Source: §VI · pp. 10–11

Practical scope

  • The efficient quantum-Fisher-information approximation has negligible error in practically relevant scenarios and supports scalability limited by gate number and quality rather than qubit number.The approximation is designed for nonunitarity caused by small imperfections in individual gates.Source: §VI · p. 11

Generality

  • The method also applies beyond noisy unitary circuits to intentional non-unitary operations such as measurements or variable-time decoherence.This flexibility is stated as a broader applicability of the density-operator formulation.Source: §VI · p. 11

Appendix A: Imaginary time evolution of mixed quantum states

Mixed-state imaginary-time evolution improves on simple gradient descent but has geometric and convergence limitations that motivate the paper’s natural-gradient approach.

  • Mixed-state imaginary-time evolution can increase or decrease the density operator’s mixedness and does not reduce to pure-state imaginary-time evolution for pure states.Source: Appendix A · p. 13
  • The method uses a parameter update with vector Y and a Hilbert-Schmidt metric M built from mixed-state differentials.Y_k replaces the energy gradient, while M contains Hilbert-Schmidt scalar products between state differentials.Source: Appendix A · p. 13
  • Although improved over simple gradient descent, the evolution can become stuck away from the optimum because its exact gradient points in a purely non-unitary direction.Source: Appendix A · p. 13
  • For pure states, M reduces to the quantum Fisher information, but Y does not reduce to the pure-state gradient when global phase evolution contributes.The pure-state expression is Y_k = −g_k/2 − E Re⟨∂_kψ|ψ⟩.Source: Appendix A · pp. 13–14
  • The mixed-state formulation differs from the paper’s quantum Fisher metric because M is Riemannian but not monotone under CPTP maps, and it need not converge to the true optimum.Source: Appendix A · p. 14

Appendix B: Analytically solvable case of single qubits

The single-qubit example analytically compares natural-gradient and simple-gradient evolution in Bloch coordinates, showing that natural gradients remove parametrisation dependence and improve convergence.

  • The single-qubit cost is E(y,z)=z, while the quantum Fisher metric produces natural-gradient vector g_n=(−yz, 1−z^2).Source: Appendix B · p. 15
  • The natural-gradient differential equations are solved analytically from arbitrary initial Bloch vectors and compared with numerically simulated simple-gradient trajectories.Source: Appendix B · p. 15
  • Simple gradient descent depends strongly on parametrisation and can become trapped at the Bloch-ball origin, where its polar-coordinate gradient vanishes.Source: Appendix B · p. 15
  • Natural-gradient evolution is parametrisation-independent, with polar-coordinate simulations matching the analytical Bloch-sphere solution.Source: Appendix B · p. 15
  • Near pure states, the metric becomes singular and its inverse slows evolution, correcting increasingly large state-space jumps caused by small coordinate steps.Source: §4 · p. 15

4. The eigenvalues of the metric tensor are (

The single-qubit comparison shows a substantially faster convergence rate for natural-gradient evolution than for simple gradient descent.

  • Natural-gradient evolution has exponential convergence, whereas simple gradient descent appears to converge only at inverse-polynomial order.Source: §4 · p. 15
  • The metric tensor becomes increasingly important for non-unitary evolution, but is constant and irrelevant when evolution is restricted to pure states in the polar parametrisation.Source: §4 · p. 15

Appendix C: Proofs

The proofs establish Fisher-information equivalence in a phase-free computational-basis setting and bound the Fourier complexity of local unitary and CPTP circuits.

  • When the state amplitudes encode a classical probability distribution, quantum and classical Fisher information coincide, with the computational basis as the optimal measurement basis.Source: Appendix C · p. 16
  • A circuit of ν L-local unitary gates has O(2^(2νL)) Fourier components, while a comparable non-unitary circuit has O(2^(4νL)) components.Source: Appendix C · p. 16
  • The Fourier expansions support objective functions E(θ)=Tr[ρ(θ)H] expressed with the same respective component-count bounds.Source: Appendix C · p. 16
  • The unitary bound follows by expanding each L-local gate into O(2^(2L)) Fourier terms and composing ν gates.Source: Appendix C · pp. 16–17
  • General L-local CPTP maps are represented through Stinespring dilation by at-most-2L-local unitaries, yielding O(2^(4L)) Fourier terms.Source: Appendix C · pp. 16–17
  • Despite purification-based proofs, the paper states that L-local non-unitary transformations can be implemented without ancillary qubits.Source: Appendix C · pp. 16–17

Appendix D: Explicit construction of effective CPTP maps

The appendix constructs effective K-local CPTP maps using randomly applied unitary transformations, then analyzes how circuit-induced mixedness affects symmetric-logarithmic-derivative relations and quantum Fisher information.

Effective CPTP-map construction

  • Arbitrary 1-local CPTP maps can be implemented without ancillary qubits by randomly applying unitary transformations during circuit executions.For a single qubit, rotations change Bloch-sphere angles while depolarization changes the Bloch-vector length.Source: Appendix D · p. 17
  • The unitary-reflection protocol using Pauli X, Y, and Z operators, including signed probabilities, fully parametrizes every 1-local CPTP map.Expectation values are summed over repeated executions with the corresponding probabilities or sign changes.Source: Appendix D · p. 17
  • An arbitrary density matrix can be decomposed into a pure state and unitary transformations with probabilities, enabling expectation values to be estimated by stochastic circuit sampling.This provides a general sampling construction beyond the single-qubit example.Source: Appendix D · p. 17

Mixed-state error analysis

  • For high-entropy circuit-error distributions with pn = O(1/d) for n > 1, ϵ-mixed states satisfy symmetric-logarithmic-derivative relations with leading error E(ϵ) = O(κ2/d) + O((1−ϵ)κ2), where d = 2N.Source: Appendix D · p. 19
  • F_Q[ρ_ϵ]_kl ≤ (1−ϵ)F_Q[ρ_ideal]_kl + O(ϵκ/d) under the same high-entropy assumption, relating noisy and ideal quantum Fisher information.Here ρ_ideal = |ψ_n⟩⟨ψ_n| and d is the system dimensionality.Source: Appendix D · pp. 19–20

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