Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows

Charles Hadfield; Sergey Bravyi; Rudy Raymond; Antonio Mezzacapo

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Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows

Charles Hadfield, Sergey Bravyi, Rudy Raymond, Antonio Mezzacapo

arXiv:2006.15788v1Jun 29, 2020quant-ph

TL;DR

Estimating molecular-Hamiltonian expectation values accurately is important for quantum algorithms, while measurement-saving approaches can increase circuit depth. The paper introduces locally-biased classical shadows optimized using the Hamiltonian and a classical reference state, and benchmarks them on molecular systems, finding consistent variance improvements without increasing circuit depth.

Problem
Precise estimation of quantum observables, including molecular Hamiltonian energies, remains a central measurement problem for quantum algorithms.
Method
The method locally optimizes per-qubit Pauli measurement distributions using the target Hamiltonian and a classical approximation called a reference state.
Results
The estimator produces consistent variance improvements over previous methods for average-value estimation that do not increase circuit depth.
Takeaways & Limitations
Locally-biased classical shadows provide a domain-specific measurement estimator benchmarked on molecular Hamiltonians up to 16 qubits.
Takeaways & Limitations
The optimization uses reference-state assumptions, including Hartree–Fock or related approximations, and the reported molecular benchmark uses a Hartree–Fock single-reference state.

Abstract

from arXiv · show

Obtaining precise estimates of quantum observables is a crucial step of variational quantum algorithms. We consider the problem of estimating expectation values of molecular Hamiltonians, obtained on states prepared on a quantum computer. We propose a novel estimator for this task, which is locally optimised with knowledge of the Hamiltonian and a classical approximation to the underlying quantum state. Our estimator is based on the concept of classical shadows of a quantum state, and has the important property of not adding to the circuit depth for the state preparation. We test its performance numerically for molecular Hamiltonians of increasing size, finding a sizable reduction in variance with respect to current measurement protocols that do not increase circuit depths.

1. Introduction

Estimating molecular Hamiltonian expectation values is central to quantum algorithms, but existing measurement-saving methods may increase circuit depth. The paper introduces locally-biased classical shadows to reduce estimator variance without increasing state-preparation depth.

Estimating observables from quantum processors is a central subroutine in quantum algorithms, including molecular-energy estimation in VQE.
Recent methods reduce measurement counts but can increase circuit depth, conflicting with VQE’s goal of keeping gate counts low.
Pauli grouping and machine-learning strategies reduce measurements without increasing circuit depth, while relying on grouping or assumptions about the target system.
Classical shadows characterize quantum states through random measurements that can later retrieve arbitrary observables.
The proposed estimator locally biases random Pauli bases using a target observable and a classical reference state, with convex optimization in certain regimes.
The paper benchmarks molecular Hamiltonians and reports consistent variance improvements over methods that do not increase circuit depth.

2. Classical Shadows Using Random Pauli Measurements

Uniform classical shadows estimate observables by randomly selecting single-qubit Pauli measurement bases and combining compatible measurement outcomes. The resulting estimator is unbiased for the desired expectation value.

Classical shadows use random Pauli measurements to estimate an observable expectation from quantum-state measurement data.
The observable is written as a linear combination of Pauli terms, and the task is to estimate tr(ρO).
A Pauli basis is selected independently of the observable, allowing nonzero estimates for Pauli operators that qubit-wise commute with the selected bases.
The protocol combines local compatibility factors across qubits to determine contributions from Pauli operators.
The estimator is unbiased: E(ν) = tr(ρO).

3. Locally-Biased Classical Shadows

Locally-biased classical shadows replace uniform basis sampling with per-qubit probability distributions while preserving unbiased estimation. Their sampling distributions can then be optimized for a target observable and reference state.

Locally-Biased Classical Shadows: Locally-biased classical shadows sample each qubit’s Pauli basis from its own distribution and retain an estimator for tr(ρO).
Locally-Biased Classical Shadows: The uniform classical-shadow protocol is recovered when every local Pauli probability equals 1/3.
Locally-Biased Classical Shadows: The estimator measures each qubit in a randomly selected local basis and combines the resulting eigenvalues using the product basis P.
Locally-Biased Classical Shadows: Algorithm 2 recovers the expectation tr(ρO), so locally biased sampling preserves unbiasedness.
Locally-Biased Classical Shadows: The required sample count scales as S = O(ε^-2 Var(ν(s))) for fixed state and observable.
Locally-Biased Classical Shadows: Variance analysis distinguishes the locally biased estimator from the uniform setting, whose state-independent bound scales with operator weight and norm.

4. Optimised Locally-Biased Classical Shadows

The section optimises locally-biased classical-shadow measurement distributions using partial classical knowledge of a reference quantum state. It develops cost functions for single- and multi-reference states, including a convex diagonal surrogate whose minimisation controls the original variance cost.

Reference-state information: The optimisation uses a classically obtained approximation of the underlying quantum state, including Hartree–Fock, fermionic Gaussian, and perturbative reference states.The same strategy is described as extendable to generic many-body Hamiltonians when suitable reference states can be generated.
Single-reference optimisation: For a single logical basis-state reference, the estimator variance is independent of the Hamiltonian’s constant term, so optimisation focuses on its traceless component.The Hartree–Fock reference is represented as a product state with single-qubit signs m_i ∈ {±1}.
Single-reference optimisation: The diagonal cost function restricts attention to diagonal influential-pair terms and provides a tractable alternative to the generally non-convex original cost function.The influential-pair constraints require each local Pauli pair either to agree or to be drawn from {I, Z}.
Single-reference optimisation: The diagonal cost function is convex, so its minimised probability distributions provide a global minimum for that cost.Convexity follows from the convexity of −log β_i(Q_i) and preservation under positive linear combinations.
Single-reference optimisation: Although derived from the maximally mixed state, the diagonal cost function is reported to give numerically satisfying results and upper-bounds the original cost function.Therefore, minimising it implies minimising the original cost function per Pauli term in the Hamiltonian.
Multi-reference optimisation: The optimisation framework is extended to multi-reference states such as fermionic Gaussian states and perturbative solutions by constructing a corresponding multi-reference cost function.The resulting cost function is built from amplitudes of the multi-reference state and the single-qubit basis components of its terms.

5. Numerical experiments on molecular Hamiltonians

The LBCS estimator is benchmarked on molecular Hamiltonians up to 16 qubits, with variance compared across estimators, encodings, and optimization cost functions. It generally outperforms the alternatives, while encoding and cost-function choices affect the observed variance.

Benchmark setup: The benchmark covers six molecular Hamiltonians represented with up to 16 qubits and evaluates LBCS on exact ground states.Hartree–Fock states provide the single-reference state used to optimize the β distributions.
Compared estimators: The comparison includes ℓ1 sampling, LDF grouping, and unbiased classical shadows alongside LBCS.Table 1 distinguishes LBCS optimized with Eq. (14) from the diagonal cost function defined in Eq. (16).
Variance comparison: LBCS outperforms the other estimators in all but one Table 1 experiment.The exception is H2 in a minimal basis, where LDF has lower variance; the authors attribute this to the small qubit count.
Encoding effects: Jordan–Wigner, parity, and Bravyi–Kitaev encodings produce different variances, with parity and Bravyi–Kitaev generally higher in the reported analysis.Their Pauli distributions contain more X and Y operators, whereas Jordan–Wigner has a Z-heavy tail that β can bias toward.
Optimization effects: The two β-optimization cost functions produce very similar variance despite the diagonal cost function being convex.The diagonal cost function is used for the optimized distributions shown in the supplied figure and Table 2 caption.
Distribution structure: For the 14-qubit H2O Jordan–Wigner example, the optimized probability distributions are shown over the first 7 qubits.The probabilities are symmetric between corresponding spin-up and spin-down qubits and symmetric in X and Y.

6. Conclusion

The paper proposes locally-biased classical shadows for molecular energy estimation and benchmarks them on systems up to 16 qubits. The method consistently improves over Pauli grouping in the reported experiments while avoiding computationally intensive grouping problems.

Method: The proposed locally-biased classical shadows require per-qubit probability distributions optimized for a molecular Hamiltonian and a reference state.The optimization is formulated as a convex problem for the reported diagonal cost function.
Results: The benchmark covers molecular systems up to 16 qubits and reports significant, consistent improvement over the LDF heuristic.The comparison notes that other grouping heuristics differ in the number of qubit-wise commuting sets by only 10%.
Computational scope: The reported improvement is obtained without solving the computationally intensive node-colouring and minimum-clique-covering problems used by Pauli grouping methods.This contrasts the computational requirements of the proposed optimization and the grouping baselines.
Novelty: The domain-specific cost function is presented as novel and sufficiently general for applications beyond quantum chemistry.The stated broader relevance is not restricted to molecular Hamiltonians.

Appendix A. Comparative Algorithms for Estimating Molecular Hamiltonians

The appendix introduces notation and baseline procedures for estimating molecular Hamiltonian energies, including ℓ1 sampling over Pauli operators and their measurement bases.

Setup: The appendix assumes an n-qubit molecular Hamiltonian H and a provided state ρ whose energy is to be estimated.The traceless part of H and the ℓ1-norm of its traceless coefficients define the baseline sampling distribution.
Measurement notation: For a Pauli operator P, measured eigenvalues on its support are multiplied to estimate tr(ρP).The notation μ(P,i) records the eigenvalue from measuring qubit i in the corresponding Pauli basis.
ℓ1 baseline: The ℓ1 algorithm samples a Pauli operator according to the coefficient distribution γ and measures the state in the Pauli basis specified by that operator.This provides an estimate of each sampled Pauli expectation value for energy estimation.

A.1. Ell-1 algorithm.

The ℓ1 algorithm estimates a molecular Hamiltonian energy by sampling Pauli operators according to coefficient weights, measuring the required qubits, and combining the resulting eigenvalues.

Measurement: Each qubit in the support of P is measured in its corresponding Pauli basis to obtain μ(P,i) ∈ {±1}.The support-restricted measurements provide the eigenvalue factors used by the estimator.
Estimator: The product of the measured eigenvalues gives an unbiased estimate of tr(ρP) for the sampled Pauli operator.The expectation is taken over measurement outcomes for fixed P and over the γ sampling distribution.
Variance: The appendix separately derives the estimator’s single-sample variance.The variance calculation treats the Pauli-selection distribution and measurement outcomes as separate sources of expectation.
Hamiltonian structure: The baseline is formulated for a Hamiltonian decomposed into K collections of Pauli terms.

A.2. Largest degree first.

The largest-degree-first heuristic decomposes nonidentity Pauli terms into qubit-wise commuting collections, which are measured using an associated full-weight Pauli operator. Sampling collections and measuring the selected operator yields an unbiased estimator of the Hamiltonian expectation value.

Measurement construction: Each qubit-wise commuting collection has an associated full-weight Pauli operator that commutes qubit-wise with every term in the collection.This operator supplies the single-qubit measurement bases used for the collection.
Measurement procedure: Algorithm 4 samples a collection from a κ-distribution, prepares the state, and measures every qubit in the selected Pauli operator basis.The procedure is titled energy estimation via decomposition into commuting terms.
Estimator: For a fixed collection, combining the expected measurement outcome of its full-weight Pauli operator with the relevant support outcome recovers each included Pauli expectation, and averaging over κ recovers tr(ρH) in expectation.The analysis assumes without loss of generality that H is traceless.
Graph construction: LDF constructs a graph whose vertices are nonzero nonidentity Pauli terms and whose edges connect terms that anticommute on at least one qubit.The vertices are then sorted by decreasing degree for greedy coloring.
Graph construction: Greedy coloring assigns Pauli terms to collections that commute qubit-wise, with at most 1 + ∆(G) collections.The smallest available color is assigned progressively to vertices in degree order.