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Application of fermionic marginal constraints to hybrid quantum algorithms

Nicholas C. Rubin, Ryan Babbush, Jarrod McClean

arXiv:1801.03524v2quant-phphysics.chem-ph

TL;DR

Near-term hybrid quantum algorithms rely on marginal tomography, but independently estimating marginal components can require many measurements for chemical accuracy. This paper applies fermionic n-representability constraints to reduce measurement costs and project corrupted marginals, reporting more than an order-of-magnitude reduction and restored physicality in a four-qubit H2 system under three error channels.

  • Problem

    Partial tomography in hybrid quantum algorithms estimates reduced-density-matrix components without exploiting their algebraic and geometric structure, while chemical-accuracy estimation can require many measurements.

  • Method

    The paper measures fermionic marginals, uses n-representability and p-positivity constraints to reduce Hamiltonian norm, and projects measured marginals with polynomial-time reconstruction procedures.

  • Results

    More than an order-of-magnitude measurement reduction is observed for linear hydrogen chains, while projection restores physicality in H2 energy curves under single-qubit error channels.

  • Takeaways & Limitations

    Fermionic marginal constraints provide techniques for improving quantum-chemistry measurements and reconstructing physical marginals on pre-fault-tolerant quantum hardware.

Abstract

from arXiv · show

Many quantum algorithms, including recently proposed hybrid classical/quantum algorithms, make use of restricted tomography of the quantum state that measures the reduced density matrices, or marginals, of the full state. The most straightforward approach to this algorithmic step estimates each component of the marginal independently without making use of the algebraic and geometric structure of the marginals. Within the field of quantum chemistry, this structure is termed the fermionic $n$-representability conditions, and is supported by a vast amount of literature on both theoretical and practical results related to their approximations. In this work, we introduce these conditions in the language of quantum computation, and utilize them to develop several techniques to accelerate and improve practical applications for quantum chemistry on quantum computers. We show that one can use fermionic $n$-representability conditions to reduce the total number of measurements required by more than an order of magnitude for medium sized systems in chemistry. We also demonstrate an efficient restoration of the physicality of energy curves for the dilation of a four qubit diatomic hydrogen system in the presence of three distinct one qubit error channels, providing evidence these techniques are useful for pre-fault tolerant quantum chemistry experiments.

INTRODUCTION

Near-term hybrid quantum algorithms require partial tomography to estimate observables, but chemical-accuracy targets can make measurement costs large. This work uses fermionic marginal structure and n-representability constraints to reduce measurements and reconstruct physical marginals.

  • Measurement challenge: Chemical-accuracy estimation requires ⟨H⟩±1.6 × 10^-3 Hartree, potentially demanding many state preparations and measurements.The tolerance is tied to matching thermochemical properties such as heats of formation and ionization potentials.
  • Measurement challenge: 1019 total measurements were estimated for ferredoxin energy simulation under grouping and coefficient-dropping strategies.Although the estimate used pessimistic assumptions, its scale motivates accelerating operator averaging.
  • Approach: Fermionic n-representability conditions provide structure for marginals that can replace independent component estimation.The work focuses on two-particle reduced density matrices for fermionic systems with at most pair-wise interactions.
  • Approach: The proposed approach reduces Hamiltonian norm and uses p-positivity constraints to project measured marginals into the set of allowed marginals.The paper presents two computational procedures for projection and examines variance reduction in linear hydrogen chains.
  • Scope: The paper applies marginal methods to quantum chemistry and discusses their use for measurement, projection, physicality restoration, and hybrid-algorithm workflows.Its structure covers marginal theory, concentration bounds, variance reduction, projection procedures, and error-channel reconstruction.
  • Marginal representation: A p-marginal is obtained by tracing out q qubits from an n-qubit state so that n − q = p.The resulting reduced-density matrices can be represented through marginalization of the joint state.
  • Marginal representation: A 2-RDM is obtained from the n-particle density matrix by integrating out particles 3 through n.The paper uses marginals and reduced-density matrices interchangeably for these reduced objects.
  • Marginal representation: Marginal normalization can use n!/(n − p)! when indices range over all modes, and this work chooses that convention for computational ease.The normalization convention depends on how fermionic indices are ordered and ranged.

II. THE n-REPRESENTABILITY PROBLEM

The n-representability problem concerns identifying physically valid reduced density matrices using efficiently usable constraints. Prior work develops approximate constraint sets and formalizes ensemble and pure-state representability structures.

  • Problem and framework: Polynomial-size p-marginals are attractive quantum-system representations, and fermionic n-representability theory supplies constraints for selecting efficient approximations.The framework derives representability conditions and then selects subsets suitable for computation.
  • Prior theory: Prior work formalized approximate n-representability through polar-cone parameterization and later characterized complete ensemble constraints and pure-state constraints.These developments provide the theoretical basis for approximate constraint systems used in fermionic marginal methods.

A. n-Representability by Characterizing the Polar Cone

The polar-cone formulation characterizes representable 2-marginals through positivity conditions, but exact checking is intractable. Polynomial-size approximate cones yield necessary constraints that can be enforced computationally.

  • Polar-cone formulation: The representable 2-marginals form a convex set on the antisymmetric two-fermion space, characterized through its polar cone.The polar cone consists of Hermitian operators satisfying a positive projection condition.
  • Polar-cone formulation: Polar-cone operators can be lifted to the n-particle space using fermionic antisymmetrization, and the bipolar theorem characterizes the 2-marginal set through the polar cone.The fermionic lift differs from the ordinary tensor-product lift used in quantum information.
  • Computational limitation: Exact polar-cone characterization requires checking infinitely many operators and positivity of an exponentially large operator.These costs motivate approximate polar-cone constructions.
  • Approximation: A polynomial-size approximate polar cone produces necessary, but approximate, representability conditions through a kth-order operator basis with k < n.Duality places the exactly representable 2-marginals inside the approximate feasible set.
  • Approximation: Restricting the operator basis to rank below two yields the standard 2-positivity conditions.These conditions are derived by limiting the rank of monomials in the approximate polar-cone basis.
  • Positivity constraints: The resulting 1D, 1Q, 2D, 2Q, and 2G matrices must be positive semidefinite and provide necessary conditions for 2-marginals.The conditions apply to mixed states as well as pure states.
  • Positivity constraints: Anticommutation-derived equalities constrain the positivity operators, and the paper uses these linear and positivity constraints to improve estimates from measured 2-RDMs.The constraints enforce fermionic algebraic relations while supporting estimation of physical quantities.

III. CONCENTRATION OF MEASURE IN p-RDMS

The section applies measure concentration to reduced density matrices and finds that random quantum states produce increasingly trivial p-particle marginals. This implies that meaningful hybrid-algorithm ansätze must impose structure rather than remain close to Haar-random states.

  • Levy’s lemma bounds deviations of Lipschitz-continuous expectation values on the quantum-state sphere using the operator’s Lipschitz constant.The expectation value of a bounded-spectrum Hermitian operator has a Lipschitz constant bounded by its norm.
  • The average 1-RDM over states with unrestricted particle number is diagonal, with entries 1/2 and average particle number M/2.
  • The average 2-RDM generalizes a diagonal matrix to an antisymmetric 4-tensor, with signs determined by fermionic antisymmetry.
  • Higher-particle RDM elements exhibit analogous concentration, with the Lipschitz constant modified by normalization factors such as 1/2 for the 2-RDM.The same concentration behavior extends to higher reduced density matrices with modified normalization-dependent Lipschitz constants.
  • Within a fixed n-particle subspace, the average 1-RDM is diagonal with equally probable occupations across the M orbitals.The derivation restricts the trace to n-particle determinants.
  • Random quantum states generate p-particle marginals that concentrate exponentially quickly toward their average, making fixed-p observables efficiently classically evaluable.The section concludes that useful hybrid ansätze must be structured to avoid trivial observables at relatively small system sizes.

IV. REDUCING OPERATOR SAMPLE VARIANCE USING n-REPRESENTABILITY CONSTRAINTS

This section formulates optimal Hamiltonian-term sampling and shows that the measurement allocation proportional to each term’s coefficient magnitude is optimal under the stated variance model.

  • The measurement problem is to choose samples M_l for each Hamiltonian term while minimizing total measurements for a target estimation accuracy.
  • The allocation M_l ∝ |w_l| is proven optimal using Lagrange conditions, rather than assumed without proof.The paper explicitly supplies the optimization argument for the choice previously suggested in the literature.
  • Under the asymptotic assumption σ_l = O(1), the derived bound confirms the optimality of choosing sample counts proportional to coefficient magnitudes.

B. Reducing Variance Using n-Representability

The paper uses equality n-representability constraints to transform Hamiltonians without changing their expectation values on representable states, reducing the sampling objective through efficient optimization. Numerical experiments show about an order-of-magnitude improvement, while broader applicability and online SDP projection remain constrained.

  • B. Reducing Variance Using n-Representability: Equality n-representability constraints relate expectation values and can be added to the Hamiltonian without changing its expectation on n-representable states.The transformed operator is isospectral to the original Hamiltonian in the fixed-particle-number sector.
  • B. Reducing Variance Using n-Representability: The variance-reduction task becomes an L1 minimization over Hamiltonian coefficients subject to mapped linear representability constraints.The Hamiltonian and constraint system are vectorized, yielding a convex optimization problem solvable by linear-programming methods.
  • B. Reducing Variance Using n-Representability: Figure 1 compares Λ2 before the transformation with eΛ2 afterward across atomic, hydrogen-ring, geometry, and active-space calculations.Blue circles denote Λ2 before applying the techniques, while orange crosses denote eΛ2 afterward.
  • B. Reducing Variance Using n-Representability: About one order of magnitude of improvement is observed for single-atom calculations in the minimal basis.
  • B. Reducing Variance Using n-Representability: The approach uses freely available OpenFermion code and linear constraints involving 1-RDM and 2-RDM traces, Hermiticity, and consistency relations.
  • B. Reducing Variance Using n-Representability: Applicability of the variance-reduction technique to QAOA and quantum spin Hamiltonians remains an open question because comparable constraint structure is generally unavailable.

V. n-REPRESENTABILITY INFORMED PROJECTION OF 2-RDMS

The section develops projection methods that map noisy or incomplete 2-RDM measurements back toward the n-representable set while balancing reconstruction accuracy and computational cost. It formulates constrained reconstruction using n-representability conditions and semidefinite programming.

  • The reconstruction strategy projects measured 2-RDMs into the n-representable set while balancing data-collection time against classical post-processing time.The section compares simple positive-semidefinite projections with methods that incorporate representability constraints.
  • Positive-semidefinite projection enforces Hermiticity, non-negativity, and fixed particle-number trace on a measured 2-RDM.The fixed-trace condition is determined by the particle number of the system.
  • Fixed-trace positive-semidefinite projection is computationally simple but may fail to restore physicality when the measured 2-RDM is sufficiently corrupted.The method does not incorporate the full set of representability conditions.
  • The reconstruction scheme minimizes the Frobenius-norm difference from known measurements subject to approximate n-representability constraints.The formulation accommodates unknown precision and potentially missing crucial 2-RDM elements.
  • The constrained reconstruction can be relaxed to a semidefinite program using linear maps between the 2-RDM and related marginals.The SDP formulation uses a Schur complement and includes trace and antisymmetry constraints.

C. Iterative Procedure for Projecting Noisy 2-RDMs Into the Approximate n-Representable Subspace

The iterative projection procedure enforces approximate 2-positivity by cycling a noisy 2-RDM through related marginals and applying positivity and trace projections. It offers an alternative to SDP projection but does not enforce every linear representability constraint.

  • The iterative method sequentially maps 2D to 2Q and 2G, enforcing positivity and trace constraints at each marginal.It was originally developed to impose approximate n-representability on response-theory 2-RDMs.
  • Unlike SDP projection, the iterative procedure is computationally faster but cannot guarantee enforcement of linear constraints preserving projected spin and total-spin expectation values.Any representable 2-RDM is a valid fixed point of the iteration.
  • The procedure is motivated as an alternative to SDP projection because high-order polynomial SDP scaling can make online or on-the-fly projections infeasible.The comparison is therefore between stronger constrained optimization and faster iterative processing.
  • The procedure checks fixed-trace and eigenvalue positivity across 2D, 2Q, and 2G to determine convergence.The iterative flow is continued until the stopping condition is satisfied.

D. Reconstruction results

The reconstruction experiments evaluate noisy 2-RDMs for hydrogen systems and then test SDP-based physicality restoration under single-qubit error channels. Representability projections reduce estimator variance while introducing bias, and symmetry-constrained reconstruction smooths energy curves but can shift their minima.

  • Reconstruction of small systems: The experiments corrupt pure-state 2-RDMs with Gaussian sampling noise and compare four projection procedures using energy and chemical observables.The systems include diatomic hydrogen and a linear four-hydrogen chain.
  • Reconstruction of small systems: n-representability projections generally reduce the energy-estimator variance while introducing bias.Figure 3 decomposes mean-squared error into variance and squared bias for the H2 energy estimator.
  • Reconstruction of small systems: SDP projection gives zero mean-squared error for Sz, S2, and particle number when those expected values are imposed as constraints.The resulting correction is described as restoration of physicality.
  • Reconstruction of marginals from the variational channel state model: The SDP reconstruction restores physicality for H2 energy curves subjected to dephasing, amplitude damping plus dephasing, and depolarizing channels.The reconstruction uses spin-adapted 2-RDM components and 2-positive n-representability conditions.
  • Reconstruction of marginals from the variational channel state model: Symmetry-constrained curves become smooth, but the energy rises beyond 1.5 Angstroms and the potential-energy minimum shifts by -0.03, 0.062, and 0.186 Angstroms for the three channels.The shifts correspond respectively to dephasing, dephasing plus relaxation, and depolarizing noise.

VI. USE CASES OF RDMS IN AUGMENTING ENERGY EXPECTATION THROUGH PERTURBATION THEORY

The paper examines how RDM-based methods can augment energy calculations and reduce measurement burdens in hybrid quantum chemistry. It also reports reconstruction and purification strategies for noisy quantum states.

  • Perturbative energy corrections: NEVPT2 can be formulated without determinant decompositions or the full wavefunction, although its direct 4-RDM requirement motivates 2-RDM cumulant approximations.The 4-RDM is expensive to estimate on quantum computers.
  • Noisy-state reconstruction: Figure 4 compares noisy H2 energy curves under amplitude-damping, dephasing, and depolarizing channels with curves reconstructed under exact n-representability conditions.The reconstruction uses markers, while solid curves show unreconstructed energies and the black curve denotes the true ground-state energy.
  • Perturbative energy corrections: Cumulant-based approximations reconstruct higher-order RDM information from the 2-RDM by neglecting irreducible three- and four-particle components.These approximations reduce energy-estimation sampling demands but introduce errors into perturbative corrections.
  • Noisy-state reconstruction: The reconstruction approach restores physicality for pure states corrupted by single-qubit error channels, while stronger 3-RDM positivity and pure-state constraints remain prospective improvements.The discussed positivity constraints do not force marginals to originate from pure states.
  • Scope: The paper concludes that representability methods warrant further testing on realistic systems, realistic error models, and efficient numerical implementations.This is presented as a direction for establishing their utility in hybrid quantum simulation.

Appendix A: Structure of RDMs

The appendix describes the block structure of RDMs and the linear relationships that connect RDM observables to chemical Hamiltonians and constrained quantities. These structures support symmetry-aware reconstruction and observable evaluation.

  • RDM structure: The 2-RDM has block structure reflecting Hamiltonian symmetries, including spin, particle number, time reversal, and spatial symmetry sectors.The SDP can block the 2-RDM according to ⟨Sz⟩, ⟨S2⟩, ⟨n⟩, and spatial symmetry groups.
  • RDM structure: Redundant spin blocks can be removed or mapped one-to-one, while corresponding Hamiltonian blocks use antisymmetrized integrals.The 2Q and 2G matrices share related block structures, although 2G has slightly less structure.
  • Observable functionals: The energy is a linear functional of the 1-RDM and 2-RDM, with one- and two-electron integral tensors supplying the coefficients.Expected Hamiltonian values therefore follow from selected RDM elements.
  • Observable functionals: Total spin, projected spin, and particle number are likewise linear functionals of the 1-RDM and 2-RDM.Their fermionic-operator expansions produce rank-4 and rank-2 polynomials whose expectations are sums over RDM elements.
  • Reconstruction constraints: The reconstruction SDP imposes trace, contraction, and mapping constraints linking particle, hole, and particle-hole RDMs.These relations arise by rearranging fermionic ladder operators.

Appendix D: Computational Implementation of the Reconstruction Problem

The reconstruction problem is implemented as a sparse semidefinite program that exploits quantum-chemical block structure and an augmented-Lagrangian boundary-point solver. The appendix also defines MSE evaluations for noisy H2 and H4 observables.

  • SDP implementation: The reconstruction SDP is sparse because each constraint involves relatively few variables compared with the full optimization problem.Augmented-Lagrangian SDP solvers have been applied efficiently to quantum chemistry and condensed matter problems.
  • SDP formulation: The primal SDP uses a symmetric positive-semidefinite matrix variable X, linear constraints A, vector b, and trace-inner-product objective data C.The associated dual variables are y and S.
  • Boundary-point solver: The boundary-point algorithm alternates solving for y, projecting a matrix into positive and negative parts, and checking inner and outer residuals.The iteration updates k and monitors δ_inner and δ_outer.
  • Boundary-point solver: Positive projection is generated by an eigenvalue decomposition that retains positive eigenvalues and their associated eigenvectors.For larger problems, the inner minimization can use conjugate gradients with a tighter stopping tolerance.
  • Error evaluation: Figures 5 and 6 evaluate MSE for energy, total spin, projected spin, and particle number over 100 samples for H2 and H4, decomposing error into variance and bias.The figures compare estimator behavior across systems and observables rather than reporting a single aggregate metric.
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