PaperlayerView exact revision on arXiv

Source-linked AI summary

Matchgate Shadows for Fermionic Quantum Simulation

Kianna Wan, William J. Huggins, Joonho Lee, Ryan Babbush

arXiv:2207.13723v4quant-phphysics.chem-ph

TL;DR

Estimating QC-AFQMC trial-state overlaps with prior Clifford shadows requires post-processing that scales exponentially with qubit count. The paper analyzes matchgate shadows, proves the discrete Clifford-matchgate ensemble matches the continuous ensemble through three moments, and develops efficient procedures for fermionic estimates, including overlaps needed for QC-AFQMC.

  • Problem

    Prior Clifford-shadow post-processing for QC-AFQMC trial-state overlaps scales exponentially with qubit count.

    Source: §I · p. 2

  • Method

    The paper analyzes shadows from continuous and discrete matchgate-circuit ensembles and develops classical post-processing procedures for fermionic quantities.

    Source: §VII · p. 43

  • Results

    The continuous and discrete matchgate ensembles have matching first three moments, and matchgate-shadow post-processing estimates each overlap sample in O(n^4) time.

    Source: §VII · p. 43

  • Takeaways & Limitations

    Matchgate shadows provide efficient estimates of local fermionic observables and global fermionic Gaussian properties, including overlaps used in AFQMC.

    Source: §VII · p. 43

Abstract

from arXiv · show

"Classical shadows" are estimators of an unknown quantum state, constructed from suitably distributed random measurements on copies of that state [Nature Physics 16, 1050-1057]. Here, we analyze classical shadows obtained using random matchgate circuits, which correspond to fermionic Gaussian unitaries. We prove that the first three moments of the Haar distribution over the continuous group of matchgate circuits are equal to those of the discrete uniform distribution over only the matchgate circuits that are also Clifford unitaries; thus, the latter forms a "matchgate 3-design." This implies that the classical shadows resulting from the two ensembles are functionally equivalent. We show how one can use these matchgate shadows to efficiently estimate inner products between an arbitrary quantum state and fermionic Gaussian states, as well as the expectation values of local fermionic operators and various other quantities, thus surpassing the capabilities of prior work. As a concrete application, this enables us to apply wavefunction constraints that control the fermion sign problem in the quantum-classical auxiliary-field quantum Monte Carlo algorithm (QC-AFQMC) [Nature 603, 416-420], without the exponential post-processing cost incurred by the original approach.

I. INTRODUCTION

The paper develops matchgate shadows to address exponential post-processing in QC-AFQMC and establishes that continuous and discrete matchgate ensembles have the same first three moments. Its introduction also reviews the classical-shadow framework and the fermionic concepts used in the analysis.

II. BACKGROUND

  • Classical shadows are unbiased state estimators built from random measurements, with efficiency depending on the unitary ensemble and observables of interest.The background section reviews this framework and its QC-AFQMC application.Source: §I · p. 2, §II · p. 3

I. INTRODUCTION

  • QC-AFQMC constraints use trial-state inner products, but prior Clifford-shadow post-processing for those estimates scales exponentially with qubit count.The paper positions its matchgate-shadow protocols as a remedy for this bottleneck.Source: §I · p. 2
  • The continuous matchgate ensemble and its discrete Clifford-matchgate subset have equal first three moments, making the latter a matchgate 3-design with functionally equivalent shadows.The paper uses the shared moments to analyze the discrete ensemble through the more explicit symmetry of the full matchgate group.Source: §I · p. 2
  • Matchgate shadows support efficient estimates of local fermionic observables, Gaussian-state fidelities, and overlaps with Slater determinants or more general pure fermionic Gaussian states.The paper also gives a general framework for estimating other fermionic quantities, without bounding variance in that broader setting.Source: §I · pp. 2–3

A. Preliminaries and notation

  • The paper introduces notation for n-qubit states and operators before reviewing the shadow framework and its use in QC-AFQMC.Readers familiar with the background can proceed to the main results after consulting the notation.Source: §II · p. 3, A. Preliminaries and notation · p. 3

1. Majorana operators

  • The notation section defines 2n Majorana operators and uses ordered products of them to describe operator subspaces, including the subspace of even operators.Majorana operators are Hermitian and satisfy anticommutation relations.Source: §1 · pp. 4–5

2. Matchgate circuits/fermionic Gaussian unitaries

  • Matchgate circuits are qubit representations of fermionic Gaussian unitaries under the Jordan-Wigner transformation, forming the matchgate group.The circuits’ generating gates implement rotations and a reflection associated with real orthogonal transformations.Source: §2 · pp. 5–6

3. Fermionic Gaussian states and Slater determinants

  • Fermionic Gaussian states include ground and thermal states of non-interacting Hamiltonians and are characterized by two-point correlations; not every Gaussian state is a Slater determinant.Their covariance matrices provide a classical description.Source: §3 · p. 6

4. Liouville representation

The section reviews classical shadows and their resource demands, then establishes that discrete Clifford matchgate circuits reproduce the first three moments of the continuous matchgate ensemble. It also describes the QC-AFQMC overlap-estimation bottleneck that the paper’s matchgate-shadow methods address.

4. Liouville representation

  • In Liouville representation, operators are normalized double kets in the Hilbert-Schmidt norm, and Majorana products form an orthogonal operator basis.The basis yields a resolution of the identity superoperator.Source: §4 · p. 7

B. Review of the classical shadows framework

  • Classical shadows estimate observable expectations by sampling a unitary, measuring in the computational basis, and classically applying the inverse measurement channel to the outcome.The resulting random operator is an unbiased estimator; quantum operations are limited to sampling, applying the unitary, and measuring.Source: B. Review of the classical shadows… · p. 8
  • Classical shadows are not generally efficient in both quantum and classical resources: estimates can have large variance, and classical post-processing can scale exponentially with system size.This work aims for polynomial quantum and classical complexity for a broad class of fermionic observables.Source: B. Review of the classical shadows… · p. 9

C. Classical shadows applied to quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC)

  • QC-AFQMC controls the fermion sign problem with constraints involving overlaps between the trial wavefunction and statistical samples; the required overlaps are costly to evaluate one at a time with Hadamard tests.The earlier Clifford-shadow approach has constant-bounded variance for the relevant estimates, but its classical post-processing appears to scale exponentially with system size.Source: C. Classical shadows applied to… · pp. 9–10
  • The paper’s matchgate-shadow techniques are intended to remove the exponentially costly classical post-processing step in future QC-AFQMC implementations.The earlier experimental implementation used Clifford shadows because the system sizes considered were sufficiently small.Source: C. Classical shadows applied to… · p. 10

A. Random matchgate circuits

  • The matchgate group corresponds to fermionic Gaussian unitaries and, ignoring global phases, is in one-to-one correspondence with O(2n).The paper studies Haar-uniform draws from this group and uniform draws from its subgroup of Clifford matchgate circuits.Source: A. Random matchgate circuits · p. 10
  • The Haar-uniform matchgate ensemble and the uniform ensemble of matchgate circuits that are also Clifford unitaries have equal first three moments, making the discrete ensemble a matchgate 3-design.Because the shadow measurement channel and estimate variances depend on the second and third twirls, the two ensembles yield the same channel and variances.Source: A. Random matchgate circuits · pp. 10–11

B. Matchgate shadows

The section characterizes matchgate-shadow samples and explains how to compute estimates and control their variance for several fermionic targets. For QC-AFQMC overlap estimation, the protocol removes an exponential classical post-processing cost.

B. Matchgate shadows

  • Estimates are unbiased when the state or observables lie in the even-operator subspace, and the protocol requires efficient sample evaluation and variance bounds.The section notes that relevant QC-AFQMC states and observables are even operators or can be made even.Source: B. Matchgate shadows · pp. 12–13

1. Local fermionic observables

  • For local fermionic observables, sample expectations are computable via Pfaffians in O(n^3), while the variance bound scales as n^{|S|/2} for constant |S|.These observables can therefore be efficiently estimated using either matchgate ensemble.Source: §1 · p. 13

2. Gaussian density matrices

  • For Gaussian-state observables, each shadow estimate is computable in at most O(n^3), with a refined variance bound of O(sqrt(n) log n).The overlap tr(ϱρ) is a fidelity when either ϱ or ρ is pure.Source: §2 · pp. 13–14

3. Overlaps with Slater determinants

  • For even-fermion Slater determinants, overlap estimation has a polynomial-time computable variance bound; using this protocol in QC-AFQMC removes exponential classical post-processing.For ζ > 0, the authors provide no asymptotic variance bound; the plotted bound suggests it is below the ζ = 0 bound.Source: §3 · p. 15

4. More general fermionic observables

The section extends matchgate-shadow post-processing to broader fermionic observables, including overlaps with arbitrary pure fermionic Gaussian states. It situates that scope against earlier and subsequent shadow approaches.

4. More general fermionic observables

  • The framework evaluates traces of operator products when each factor is a Majorana linear combination, fermionic Gaussian unitary, or fermionic Gaussian density operator.It uses a Grassmann-integral representation and an algorithm to evaluate integrals of the resulting form.Source: §4 · p. 16
  • The framework extends overlap estimation from Slater determinants to arbitrary pure fermionic Gaussian states.The authors describe this as extending the earlier Slater-determinant overlap results.Source: §4 · p. 16

1. Prior work

  • Compared with Ref. 7, this work efficiently estimates more observable families beyond single products of Majorana operators, including |φ⟩⟨0| operators for QC-AFQMC overlap estimates.Source: §1 · p. 17

2. Subsequent work

  • For Slater-determinant overlaps, Ref. 16's O(1) average variance over all ζ-RDMs does not establish overlap variance; its protocol also adds ζ ancillary fermionic modes.The authors say their Eq. (43) gives a worst-case overlap-variance guarantee, though it is likely not independent of n; Ref. 16's ancillary modes may be prohibitive near half-filling.Source: §2 · p. 18

IV. ENSEMBLES OF MATCHGATE CIRCUITS

  • The section analyzes classical shadows from both the continuous matchgate group and its discrete matchgate-Clifford subgroup.The two distributions are defined in Section III, and the section discusses sampling and implementing their unitaries.Source: §IV · p. 18

A. Moments of the distributions

The authors evaluate twirl channels for matchgate distributions, beginning with the two-fold twirl. Symmetry constrains which Majorana-basis terms can survive that channel.

A. Moments of the distributions

  • The proof calculates the two-fold twirl directly, then uses related ideas to sketch the more technical three-fold-twirl proof.The two-fold calculation can also be derived from the three-fold twirl, but is presented as self-contained.Source: A. Moments of the distributions · pp. 19–20

1. The 2-fold twirl E(2)

  • The two-fold twirl is an orthogonal projector whose image is characterized using the Majorana-product basis.The proof first uses symmetry to exclude basis states and characterize the remaining action.Source: §1 · p. 20
  • For the two-fold twirl, a basis input survives only when S1 = S2, and diagonal inputs with equal |S| have the same output.These conditions are the stated conclusions of Lemma 1.Source: §1 · p. 20

2. The 1-fold twirl E(1)

The one-fold twirl retains only the identity component, while the three-fold twirl has a structured surviving subspace. Equality of the first three twirls makes the two matchgate shadow ensembles equivalent for their measurement channel and variances.

2. The 1-fold twirl E(1)

  • The one-fold twirl projects an operator onto its identity component in the Majorana basis.The stated channel is E(1) = |γ∅⟫⟪γ∅|.Source: §2 · p. 21

3. The 3-fold twirl E(3)

  • Three-fold-twirl basis terms can survive only when their index sets have the form S1 = A1 ∪ A2, S2 = A2 ∪ A3, and S3 = A3 ∪ A1 for mutually disjoint sets.The output is also invariant across such disjoint-set triples with matching set sizes.Source: §3 · pp. 21–22

B. Classical shadows via matchgate circuits

  • The continuous and discrete matchgate ensembles have identical measurement channels and estimator variances because their first three twirl channels coincide.The paper states this follows because the channel and variances depend on the distribution through the two- and three-fold twirls.Source: B. Classical shadows via matchgate… · p. 22

1. Measurement channel

  • The measurement channel maps onto even operators, so estimates are unbiased when the state is even or all observables are even.The channel inverse is defined on the even subspace; the alternative condition follows from Hilbert-Schmidt orthogonality.Source: §1 · p. 24

2. Variance

The paper derives variance bounds for even observables and develops efficient post-processing for Gaussian-state fidelities and overlaps. The fidelity estimator can be evaluated in time cubic in the covariance-matrix rank.

2. Variance

  • The variance bound is derived for even observables, and an arbitrary observable can be replaced by its even projection when the state is even.The shadow samples themselves are even operators, so the variance agrees with that of the observable's even projection.Source: §2 · p. 24

A. Estimating fidelities with fermionic Gaussian states

  • Theorem 2 provides an expression for evaluating a Gaussian-state fidelity from a matchgate-shadow sample.The resulting estimate is unbiased for the fidelity between the unknown state and the Gaussian state.Source: A. Estimating fidelities with… · p. 26
  • For a Gaussian state with covariance matrix of rank r, each matchgate-shadow fidelity estimate can be computed in O(r^3) time.The procedure finds polynomial coefficients using Pfaffian calculations.Source: A. Estimating fidelities with… · pp. 26–27
  • Setting z = 1 in Theorem 2 yields a general formula for the overlap between two arbitrary Gaussian states.Source: A. Estimating fidelities with… · p. 30

B. Estimating overlaps with Slater determinants

Theorem 3 gives a method for efficiently estimating the expectation value of |φ⟩⟨0| for an arbitrary Slater determinant, which enables overlap estimation.

B. Estimating overlaps with Slater determinants

C. Efficient estimation of more general observables

The section extends matchgate-shadow estimation beyond Gaussian density operators and Slater-determinant overlaps, using a general method for evaluating certain Grassmann integrals.

C. Efficient estimation of more general observables

  • The method estimates expectation values for products that include Gaussian density operators, Gaussian unitaries, and linear combinations of Majorana operators.The section describes these as a broader class of observables than the Gaussian density operators and |φ⟩⟨0| operators treated earlier.Source: C. Efficient estimation of more… · p. 32
  • This class includes |ϕ⟩⟨0| for any pure Gaussian state, allowing efficient estimation of overlaps between any pure state and any pure Gaussian state.The pure state is accessed via a state-preparation circuit.Source: C. Efficient estimation of more… · p. 32
  • The approach expresses operator traces as Grassmann integrals and efficiently evaluates integrals of the required form.The authors generalize prior results and extend a method for evaluating a certain form of Grassmann integral.Source: C. Efficient estimation of more… · p. 32

1. Review of Grassmann algebra and Grassmann integration

The section develops Grassmann-algebra tools to express operator traces as integrals, then gives an efficient algorithm for evaluating the forms needed for matchgate-shadow estimates.

1. Review of Grassmann algebra and Grassmann integration

  • Grassmann algebras use associative multiplication that is antisymmetric under exchanging generators, and the paper represents operators using Grassmann variables.The Grassmann representation preserves operator multiplication as a product only when the operator product equals the Clifford-algebra wedge product.Source: §1 · pp. 33–34

2. Expressing the trace of any product of operators as a Grassmann integral

  • Theorem 4 translates the trace of an even number of arbitrary operators into a Grassmann integral with separate generators for each operator.Independent generators preserve terms that would be lost under antisymmetric wedge multiplication with shared generators; a parity-operator case can simplify the expression.Source: §2 · p. 34

3. Efficiently evaluating certain Grassmann integrals

  • The integral-evaluation algorithm covers the required forms for products of Majorana combinations, Gaussian density operators, and Gaussian unitaries.Its runtime is O(N^3) when M is invertible and at most O(N^4) for general M, including the non-invertible case.Source: §3 · pp. 35–37

4. Worked example: Overlaps with pure Gaussian states via matchgate shadows

The worked example computes pure-Gaussian overlap quantities from matchgate-shadow samples by converting traces to Grassmann integrals and evaluating them with Algorithm 2.

4. Worked example: Overlaps with pure Gaussian states via matchgate shadows

  • The example evaluates overlaps between a pure state and an arbitrary pure Gaussian state by evaluating an expectation involving |ϕ⟩⟨0|.The pure state is accessed through a state-preparation circuit, and the example also treats a more general operator involving Majorana operators.Source: §4 · p. 37
  • The setup assumes the Gaussian state is specified as a Gaussian unitary acting on the vacuum, with the same parity as the vacuum.Source: §4 · p. 38
  • The method expresses the relevant trace as a Grassmann integral and evaluates it using Algorithm 2 after constructing matrices B and M.For a fixed z, the integral can be evaluated in at most O(n^4) time.Source: §4 · pp. 38–39
  • The estimate uses polynomial interpolation to recover coefficients of p(z), which has degree at most n, from evaluations at n + 1 values of z.Source: §4 · pp. 38–39

VI. VARIANCE BOUNDS

The section bounds estimator variances for Gaussian density operators and overlaps with Slater determinants. The Gaussian-state bound scales as O(√n log n), while the plotted Slater-determinant bounds suggest sublinear scaling.

VI. VARIANCE BOUNDS

  • Variance analysis covers Gaussian density operators and Slater-determinant overlaps, but leaves the more general observables of subsection V C to future work.The section's stated scope is the variance of estimates introduced in subsections V A and V B.Source: §VI · p. 39

A. Variance for expectation values of Gaussian density operators

  • O(√n log n) bounds the variance for estimating the expectation value of any Gaussian density operator.The bound is efficiently computable; the main-text analysis also gives a looser O(n^3) bound.Source: A. Variance for expectation values of… · p. 39

B. Variance for overlaps with Slater determinants

  • The variance bounds for overlaps with Slater determinants are efficiently computable, and their plotted growth suggests sublinear scaling for all n.For ζ > 0, the plotted bounds seem lower than the ζ = 0 bound; the sublinear behavior is suggested by the plot, not established as a proof here.Source: B. Variance for overlaps with Slater… · p. 42

VII. CONCLUSION

The conclusion presents matchgate shadows from continuous and discrete circuit ensembles, including a 3-design result for Clifford matchgates. It describes efficient estimation applications and identifies variance, optimization, and robustness questions left open.

VII. CONCLUSION

  • The first three moments of uniform matchgate and Clifford-matchgate distributions match, establishing that Clifford matchgate circuits form a matchgate 3-design.The 3-design property is used to bound estimator variance for general classes of fermionic observables.Source: §VII · p. 43
  • Matchgate shadows efficiently estimate local fermionic operators, Gaussian-state expectations, and overlaps with Slater determinants, with variance bounds in some cases.For local operators, the method matches prior work [7] and extends to arbitrary single-particle bases.Source: §VII · p. 43
  • For Gaussian density operators, shadow estimates are computable in cubic time and have variance bounded by O(√n log n).For pure Gaussian states, the expectation value gives fidelity with an arbitrary unknown quantum state.Source: §VII · p. 43
  • Matchgate shadows handle local fermionic observables and global properties together, unlike the cited Clifford-shadow choices.The comparison is between randomizing over the matchgate-Clifford subset and choosing single-qubit or n-qubit Clifford circuits.Source: §VII · p. 43
  • Matchgate shadows remove the exponential post-processing bottleneck for estimating arbitrary-state overlaps with Slater determinants, processing each sample in O(n^4) time.The overlap variance bound is efficiently evaluable, and its plotted growth suggests sublinear scaling for the shown values up to n = 1000.Source: §VII · p. 43
  • The framework extends overlap estimation to arbitrary pure fermionic Gaussian states and evaluates products of commonly encountered fermionic operators.The authors expect this framework to support further applications.Source: §VII · p. 43
  • Open questions include reducing overlap-estimation runtime, improving other post-processing methods, characterizing variance more intuitively, and handling nonlinear quantities.The authors also ask whether a robust matchgate-shadow protocol can be designed, and whether a 3-design result holds for the parity-conserving subgroup.Source: §VII · pp. 43–44

Appendix A: Reducing overlap estimation to expectation value estimation

The appendices reduce overlap estimation to matchgate-shadow expectation estimation and extend the procedure to arbitrary pure Gaussian states. They also describe sampling and compiling the continuous and discrete matchgate ensembles and provide proof details for the third-moment result.

Appendix A: Reducing overlap estimation to expectation value estimation

  • Overlap estimation is reduced to estimating an observable's expectation value, with state choices satisfying overlap, orthogonality, and even-operator conditions.The reduction uses the initial state and an observable of the form |Φ⟩⟨⊥| within the matchgate-shadow protocol.Source: Appendix A · p. 44

1. Overlaps with arbitrary Slater determinants

  • Ancilla extensions handle arbitrary Slater-determinant overlaps, including odd particle number and nonzero vacuum overlap.For odd ζ, one ancilla is introduced; for even ζ, two are introduced, and the resulting even-fermion Slater determinant permits applying the shadow protocol.Source: §1 · p. 45

2. Overlaps with arbitrary pure Gaussian states

  • The reduction extends to arbitrary pure Gaussian states by choosing ancilla states according to the Gaussian state's parity.The resulting Gaussian observables can be evaluated with the method described in subsection V C 4.Source: §2 · p. 45

1. Sampling from the uniform distribution over Mn

  • Haar-random continuous matchgates can be sampled through O(2n) and compiled using O(n^2) two-qubit gates at depth O(n).The sampling description uses Gram-Schmidt orthogonalization of a real random matrix to obtain a Haar-random orthogonal matrix.Source: §1 · pp. 45–46

2. Sampling from the uniform distribution over Mn ∩Cln

  • Discrete Clifford matchgates are sampled from signed permutation matrices, then compiled using matchgate or Clifford-circuit strategies.The discrete ensemble corresponds to signed permutation matrices in the orthogonal representation.Source: §2 · p. 46

Appendix C: Proof details for Theorem 1(iii)

  • The proof details characterize when the third-moment twirl is nonzero: each index must occur in zero or two of the three subsets.In that case, the subsets have the form S1 = A1 ∪ A2, S2 = A2 ∪ A3, and S3 = A3 ∪ A1 for mutually disjoint subsets.Source: Appendix C · p. 46
  • A self-inverse pairing of contributing subset triplets makes the derivative terms cancel in the proof details.The paired cases establish the derivative condition used in the third-moment proof.Source: Appendix C · p. 49

Appendix D: Computing Pfaffians of linear matrix functions in cubic time

Appendix D gives a cubic-time procedure for computing coefficients of a Pfaffian of a linear matrix function when its constant matrix is invertible. Appendix E supplies Grassmann-integral identities used in proofs for estimating more general observables.

Appendix D: Computing Pfaffians of linear matrix functions in cubic time

  • O(r^3) time computes all coefficients of pf(A(z)) for A(z)=B+zC when B is invertible, improving on O(r^4) polynomial interpolation.The recurrence computes derivatives at z=0; iteratively computing binomial coefficients keeps the total runtime O(r^3).Source: Appendix D · pp. 49–50
  • The procedure applies to the polynomial pϱ1,ϱ2(z) for arbitrary Gaussian states ϱ1 and ϱ2 because ϱ1 is invertible by construction.Source: Appendix D · p. 49

Appendix E: Proofs for the efficient estimation of more general observables

  • Grassmann-integral identities express exponential integrals through Pfaffians and support completing the square for terms coupling Grassmann variables.The completing-square step shifts χ by M^-1B^Tη and produces a quadratic term in η.Source: Appendix E · p. 50

1. Proof of Theorem 4

Theorem 4 is proved by reducing arbitrary operators to products of Majorana operators and matching the resulting signs in a Grassmann-integral representation. The proof also treats the parity operator as a special case.

1. Proof of Theorem 4

  • Linearity of the Grassmann representation and the spanning property of Majorana products reduce the identity to products of Majorana operators.Source: §1 · p. 50
  • Reordering anticommuting Majorana operators yields a sign, and the Grassmann-integral expression reproduces it through the even-cardinality Pfaffian sign identity.Source: §1 · pp. 51–52
  • When one operator is parity, the formula uses only the remaining m−1 sets of Grassmann variables and generalizes Equation 144 of Ref. 18 to arbitrary even m.Source: §1 · p. 52

2. Proof of correctness for Algorithm 2

The appendix proves Algorithm 2 correct by handling zero, determinant, invertible, and rank-reduction cases. It also tightens the variance bound for Gaussian-state expectation estimates to O(√n log n), compared with the main text's O(n^3) bound.

2. Proof of correctness for Algorithm 2

  • Algorithm 2 returns zero when K is odd or K>2N, and returns det(B) when K=2N.Source: §2 · p. 53
  • For invertible M, the Grassmann integral evaluates to a Pfaffian product, matching the expression returned in Step 3a.Source: §2 · p. 53
  • For noninvertible M, reducing to its rank yields a smaller problem involving B′M′^-1B′T, which has size K×K.Source: §2 · p. 54

Appendix F: O(√n log n) variance bound for expectation values of fermionic Gaussian density operators

  • O(√n log n) asymptotically bounds the variance expression b(n), tighter than the main-text O(n^3) bound for estimating tr(ϱρ).Source: Appendix F · p. 54

Related papers

Loading 2207.13723v4…