Fermionic partial tomography via classical shadows

Andrew Zhao; Nicholas C. Rubin; Akimasa Miyake

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Fermionic partial tomography via classical shadows

Andrew Zhao, Nicholas C. Rubin, Akimasa Miyake

arXiv:2010.16094v3Oct 30, 2020quant-ph

TL;DR

The paper develops a fermionic classical-shadows channel for estimating reduced-density-matrix observables and derives its shadow-norm and performance guarantees. Numerical and hardware-dependent analyses compare randomized measurement costs with deterministic strategies, while Hamiltonian averaging highlights further optimization opportunities.

Problem
Estimating a single observable requires accounting for term coefficients and covariances between simultaneously measured reduced-density-matrix elements.
Method
The method uses permutation-based fermionic Gaussian unitaries whose Majorana operators form the eigenbasis of the measurement channel, with shadow-norm guarantees derived from subgroup and frame-bound analysis.
Results
For realistic hardware parameters, reprogramming overhead produces no qualitative differences from the main-text results, while randomization typically outperforms competing methods below approximately 10^5 samples.
Takeaways & Limitations
Targeted variance optimization could substantially improve the classical-shadows ensemble, although the reported Hamiltonian-averaging comparisons assume noiseless devices.
Takeaways & Limitations
The Hamiltonian-averaging calculations assume the noiseless case.

Abstract

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We propose a tomographic protocol for estimating any $ k $-body reduced density matrix ($ k $-RDM) of an $ n $-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all $ k $-RDM elements to additive precision $ \varepsilon $ requires on the order of $ \binom{n}{k} k^{3/2} \log(n) / \varepsilon^2 $ repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for $ k \geq 2 $, as compared to prior deterministic strategies. We also adapt our method to particle-number symmetry, wherein the additional circuit depth may be halved at the cost of roughly 2-5 times more repetitions.

Appendix B: Computations with the fermionic Gaussian Clifford ensemble

The appendix develops the notation and representation theory used to analyze fermionic Gaussian Clifford measurements. It specifies the relevant Majorana spaces, parity restriction, and permutation-based group representations.

Appendix B: Computations with the fermionic Gaussian Clifford ensemble: Even-degree fermionic observables form the physical operator algebra because of the parity superselection rule.Only even operators can have nonzero diagonal matrix elements after parity-preserving fermionic evolution.
Appendix B: Computations with the fermionic Gaussian Clifford ensemble: The fermionic Gaussian unitary group is represented through orthogonal transformations and their induced action on Majorana operators.The induced map is an orthogonal representation from SO(2n) into SO(4^n−1).
Appendix B: Computations with the fermionic Gaussian Clifford ensemble: Signed permutation matrices provide the relevant generalized permutation notation, including the determinant-one subgroup used in the analysis.The appendix distinguishes symmetric, alternating, signed, and determinant-constrained permutation groups.
Appendix B: Computations with the fermionic Gaussian Clifford ensemble: The analysis begins by deriving the measurement channel for fermionic Gaussian unitaries and relating its eigenvalues to the ensemble’s shadow norm.The derivation proceeds through the channel, its eigenvalues, explicit shadows, and arbitrary-observable shadow norms.

1. The classical shadows channel

This section constructs the fermionic Gaussian Clifford shadow channel in the Majorana basis. It identifies diagonal Majorana operators, derives the channel’s permutation structure, and proves that Majorana operators are its eigenbasis.

1. The classical shadows channel: The channel is evaluated on the even fermionic operator algebra, whose natural basis is given by even-degree Majorana operators.The analysis uses linearity and evaluates the channel on a Majorana basis.
1. The classical shadows channel: Diagonal Majorana operators are those with nonzero computational-basis diagonal elements and correspond to products of occupation-number operators.Their indexed subsets are denoted D2n,2k, with explicit Jordan–Wigner examples for degrees 2, 4, and 6.
1. The classical shadows channel: Fermionic Gaussian evolution acts on Majorana operators through the induced orthogonal representation associated with the unitary’s underlying transformation.The Born-rule probabilities are evaluated using preservation of inverses under the group homomorphism.
1. The classical shadows channel: Majorana operators are traceless and Hilbert–Schmidt orthogonal, properties used to simplify the channel calculation.The derivation also separates diagonal from nondiagonal Majorana operators in the computational basis.
1. The classical shadows channel: For generalized permutation matrices, each indexed column subset has exactly one row subset with a nonzero subdeterminant.This uniqueness makes the induced representation permutation-like and yields the channel’s diagonalization.
1. The classical shadows channel: The Majorana operators are the eigenbasis of the fermionic Gaussian Clifford measurement channel.The result follows by applying the unique-subdeterminant property to the channel expression.

2. The shadow norm

The shadow norm is controlled using finite-frame bounds for the induced Majorana representation. Tight frames and irreducible permutation-group actions give optimal sample complexity within the analyzed ensemble.

2. The shadow norm: Finite-frame theory converts bounds on the induced representation into bounds on the shadow norm.The relevant frame vectors arise from the action of the representation on diagonal Majorana basis vectors.
2. The shadow norm: The shadow-norm bounds are saturated exactly when the cumulative frame is tight.The frame construction includes an orbit scaled by |G|^-1/2.
2. The shadow norm: Tight frames generated by irreducible finite-group representations provide optimal sample complexity in the shadow norm.The representation-theoretic characterization links irreducible orbits to tight frames.
2. The shadow norm: For signed permutation groups, the orbit of any basis vector spans the representation space, so every nonzero vector generates a spanning orbit.Signs do not affect the span, allowing the analysis to treat basis-vector orbits uniformly.
2. The shadow norm: The behavior of tight frames is uniform across all Majorana-index basis vectors, despite mapping-dependent tuple labels.The analysis therefore does not require selecting a special diagonal Majorana operator.
2. The shadow norm: All irreducible induced representations yield the same frame bound, avoiding exhaustive subgroup search for the best ensemble.The equality follows from the explicit frame operator for irreducible group orbits.
2. The shadow norm: The theorem and its asymptotic form follow by combining the frame-based shadow-norm analysis with classical-shadow guarantees and Stirling’s approximation.The appendix states the resulting theorem for irreducible subgroups and then simplifies the expression asymptotically.
2. The shadow norm: The final corollary supplies the shadow-norm guarantee for all irreducible subgroups of the signed-permutation group.The result is stated for every relevant Majorana index and then used to derive the main-text theorem.

4. Variance bounds for arbitrary observables

This section extends the analysis from individual Majorana observables to arbitrary fermionic observables. It bounds estimator variance using the observable’s Majorana coefficients and shadow norm.

4. Variance bounds for arbitrary observables: The observable is expanded in even-degree Majorana operators, with coefficients h_µ and no identity contribution for variance calculations.For an operator of degree at most k, coefficients with |µ| > 2k are set to zero.
4. Variance bounds for arbitrary observables: Most physical observables considered here satisfy k ≤ 4, with k = 2 particularly relevant for electron–electron interactions.This provides the stated practical scope for the arbitrary-observable bound.
4. Variance bounds for arbitrary observables: The triangle inequality for the shadow norm yields an upper bound on the estimator variance for tr(Oρ).The bound is obtained by combining the observable’s Majorana expansion with the individual shadow norms.
4. Variance bounds for arbitrary observables: The asymptotic variance estimate is obtained by further loosening the exact bound.The resulting scaling is summarized for the relevant observable-degree regime.

5. Performance guarantees without median-of-means estimation

The section establishes that ordinary sample means can attain the same rigorous sampling guarantees as median-of-means estimators when classical-shadow estimators are bounded. This condition holds for the fermionic ensembles considered when estimating Majorana operators.

The sampling bound follows by applying Bernstein’s inequality to independently obtained classical shadows with bounded estimators.
Bounded classical-shadow estimators allow sample means to estimate all observables simultaneously with probability at least 1 − δ.The guarantee uses a union bound over L observables and Bernstein’s inequality.
The sample-mean estimator requires no median-of-means procedure for the ensembles presented with Majorana observables.The boundedness condition applies to UFGU and UNC.
For small ε, the factor (1 + ε/3) is approximately 1, simplifying the practical sampling requirement.
The resulting bound has smaller numerical factors than the cited median-of-means analysis, although that comparison is not claimed to reflect an optimized prior proof.

2. Universal upper bounds on the shadow norm

The section bounds the shadow norm by analyzing Majorana-operator locality under randomized fermionic permutations, using Jordan–Wigner locality as a universal upper bound. Combinatorial counting yields an n-independent hypergeometric factor for constant k.

Jordan–Wigner locality provides a universal upper bound because the mapping is maximally nonlocal.
The maximum locality of a 2k-degree Majorana operator lies between 2k and n under Jordan–Wigner encoding.
The locality distribution is counted by enumerating index configurations, permutations, and placements of trivial indices using stars-and-bars arguments.
The locality sum can be expressed through the Gauss hypergeometric function 2F1.
For positive integers n and constant k, 1 ≤ 2F1(k, 2k − n; k − n; 1/3) ≤ (3/2)^k, so this factor is independent of n.
The resulting lower-bound analysis gives the shadow norm scaling for k = O(1).

1. The 1-RDM method

The 1-RDM method uses fermionic swap circuits to relabel orbitals so off-diagonal elements become local qubit observables, followed by parallel Pauli measurements. The construction requires 4⌈n/2⌉ + 1 measurement circuits.

The method measures real and imaginary off-diagonal 1-RDM components using Pauli-basis observables.
Fermionic swaps relabel orbitals so off-diagonal 1-RDM terms avoid highly nonlocal Jordan–Wigner strings.
The swap circuits are Gaussian and number-preserving, and products of fermionic swaps can be consolidated into depth-n circuits.
⌈n/2⌉ swap circuits suffice to make every orbital pair nearest-neighbor at least once.The circuits can be implemented using a parallelized odd–even transposition sort.
4⌈n/2⌉ + 1 measurement circuits are required after accounting for four Pauli bases per permutation.

2. The 2-RDM method

The 2-RDM method extends fermionic swap networks to arrange disjoint orbital pairs for measuring 4-local terms. Its general construction uses combinatorial pair coverings, while quantum overlap tomography supplies parallel measurements with polylogarithmic overheads.

The 2-RDM construction swaps orbital orderings into combinations of nearest-neighbor pairs to measure general 4-local terms.
The number of required swap circuits follows from counting disjoint combinations of nearest-neighbor pairs.
For n > 7, the 16 Pauli operators cannot be measured in 16 bases, so quantum overlap tomography is used instead.
Quantum overlap tomography requires almost all relevant 4-qubit marginals and provides an O(log(n)) bound when a perfect hash family is available.
The explicit quantum overlap tomography construction works for any n but is not asymptotically optimal.
The general k-RDM construction has a lower bound of 4^k unique measurement circuits, with polylogarithmic factors arising from parallel Pauli measurements.

a. Upper bounds for k = 3, 4

The appendix constructs upper bounds for measuring 3- and 4-RDM elements by partitioning terms according to their numbers of unique indices. The swap-network EQOT protocol is compared with naive measurement scaling as a function of fermionic modes.

3-RDM: The 3-RDM is partitioned into terms with 3, 4, 5, and 6 unique indices before counting measurement circuits.Each index class is handled using measurement constructions tailored to its locality and operator structure.
3-RDM: Terms with three unique indices are measured using a single qubit permutation, while higher-index terms use swap circuits and EQOT on marginal operators.The six-index terms require separate treatment of imaginary components and six-qubit marginal measurements.
3-RDM: The 3-RDM circuit-count bound is deliberately loose because the marginal-measurement bound is itself an upper bound and includes unnecessary marginal terms.The appendix explicitly notes that not all marginal terms need to be measured.
Scaling comparison: Figure 2 compares swap-network EQOT scaling with naive measurement of the unique upper triangle of the k-RDM supermatrix as the number of modes increases.The swap-protocol upper bound has quadratic improvement over naive scaling.
4-RDM: The 4-RDM is similarly partitioned into terms with 4, 5, 6, 7, and 8 unique indices to obtain an upper bound on measurement configurations.The resulting accounting extends the same unique-index strategy used for the 3-RDM.

Appendix E: Supplementary numerical calculations

The supplementary numerical section examines subtler behavior of the partial-tomography scheme beyond its primary results.

Supplementary numerical calculations: The numerical studies explore additional aspects of the partial-tomography scheme that are not essential to the primary results.These calculations focus on subtler points of the method’s behavior.

1. Hyperparameter tuning

The hyperparameter r controls how many randomly generated circuits cover every observable, and increasing r reduces uneven coverage caused by outlier events.

Hyperparameter tuning: The hyperparameter r requires every observable to be measured at least r times across the generated unitaries.The total number of unitaries is denoted K_r.
Hyperparameter tuning: K_r/r decreases as r increases because larger samples reduce outlier events in which some observables are covered more often than others.This behavior is demonstrated numerically for the 2-RDM and is consistent with classical-shadow randomization.

2. Realistic time estimates

The supplementary time analysis evaluates randomized and deterministic schemes under realistic device-loading and sampling assumptions. It finds no qualitative change to the main results and identifies the sampling regime where randomized methods gain an advantage.

Circuit-loading overhead: Randomized schemes generally require more unique circuits than deterministic clique covers, so circuit reprogramming can contribute nontrivial wall-clock overhead.The analysis tests whether this overhead changes the main conclusions.
Circuit-loading overhead: The time model assumes a fixed-circuit sampling rate f_samp and a loading time t_load for each new circuit.These parameters determine the total measurement time for the two paradigms.
Numerical coverage: Figure 3 averages K_r/r over 10 random circuit collections for each r and reports one-standard-deviation uncertainty bars.The figure concerns coverage of 2-RDM observables under the UFGU ensemble.
Realistic device estimates: For Google Sycamore parameters f_samp = 5 × 10^3 Hz, t_load = 0.1 s, and S = 2.5 × 10^5, the time analysis shows no qualitative differences from the main-text results.The chosen S matches the number of shots used to estimate 1-RDM elements in a recent Hartree–Fock experiment.
Precision dependence: The randomized-method advantage typically begins below S ∼ 10^5, although the threshold varies with n and k.When S ≫ f_samp t_load, randomized measurement time scales linearly with S.

3. Hamiltonian averaging

Hamiltonian averaging estimates a single observable by accounting for term coefficients and covariances, then comparing measurement strategies through an equivalent variance framework. The numerical analysis uses molecular Hamiltonians and finds that classical shadows have room for improvement, while optimization-based and noise-aware strategies offer practical trade-offs.

Motivation: Single-observable repetition costs depend on term coefficients and covariances between simultaneously measured RDM elements.The corresponding requirement can be calculated from the single-shot variance of the estimator.
Numerical setup: The numerical studies use molecular Hamiltonians at equilibrium geometry, with k = 2 and specified minimal orbital basis sets.The Hamiltonians were generated through OpenFermion and Psi4; H2 used 6-31G, while other molecules used STO-3G.
Strategy comparison: The comparison evaluates CS (FGU), basis-rotation grouping, locally biased classical shadows, and standard Pauli classical shadows through estimator variances.Table I reports variances in Ha^2, using the exact ground state as the reference state.
Comparison framework: Deterministic measurement scheduling is reframed as randomized allocation so that an equivalent variance quantity can be computed across methods.The decomposition into measurable terms and measurement probabilities determines the resulting estimator variance.
Comparison framework: The analysis does not require randomization of the deterministic strategy; it only normalizes measurement allocations to produce an equivalent figure of merit.This qualification limits the interpretation of the deterministic-versus-randomized comparison.
Assumptions and implications: The optimal measurement distribution depends on the unknown state, so state-independent or classically tractable approximations may be used instead.For Table I, the analysis uses the exact ground state within the model chemistry.
Results and limitations: The classical-shadows approach has substantial room for improvement, while optimization-based allocation is more efficient in the reported noiseless Hamiltonian-averaging comparison.The cited discussion notes that noise resilience and particle-number postselection can reduce noise-induced variance contributions for another strategy.