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Variational Quantum State Eigensolver

M. Cerezo, Kunal Sharma, Andrew Arrasmith, Patrick J. Coles

arXiv:2004.01372v2quant-ph

TL;DR

The paper addresses extracting eigenvalues and eigenvectors from density matrices, extending variational eigensolver methods beyond Hamiltonians. It introduces VQSE, which uses majorization-based costs and one copy of the state per iteration, and demonstrates PCA and error-mitigation applications.

  • Problem

    Near-term quantum computers need methods for extracting spectra and eigenvectors of very large matrices, while VQE specifically addresses Hamiltonians rather than density matrices.

  • Method

    VQSE variationally estimates the largest eigenvalues and corresponding eigenvectors of ρ using a majorization-based Hamiltonian cost function and one copy of ρ per iteration.

  • Results

    VQSE numerically supported principal-component analysis and error mitigation, with adaptive costs producing smaller PCA errors and re-purification improving fidelity with a target W state.

  • Takeaways & Limitations

    VQSE provides a qubit-frugal approach for estimating density-matrix eigensystems and can support PCA, entanglement spectroscopy, and noise reduction.

  • Takeaways & Limitations

    For PCA, the paper notes that VQSE yields high accuracy for low-rank states.

Abstract

from arXiv · show

Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of near-term quantum computers. The Variational Quantum Eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we address the case when the matrix is a density matrix $ρ$. We introduce the Variational Quantum State Eigensolver (VQSE), which is analogous to VQE in that it variationally learns the largest eigenvalues of $ρ$ as well as a gate sequence $V$ that prepares the corresponding eigenvectors. VQSE exploits the connection between diagonalization and majorization to define a cost function $C=\Tr(\tildeρ H)$ where $H$ is a non-degenerate Hamiltonian. Due to Schur-concavity, $C$ is minimized when $\tildeρ = VρV^\dagger$ is diagonal in the eigenbasis of $H$. VQSE only requires a single copy of $ρ$ (only $n$ qubits) per iteration of the VQSE algorithm, making it amenable for near-term implementation. We heuristically demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.

I. INTRODUCTION

VQSE extends variational eigensolvers from Hamiltonians to density matrices, estimating leading eigenvalues and preparing corresponding eigenvectors with a single copy of ρ per iteration. It uses a majorization-based cost function and adaptive Hamiltonian training for near-term applications.

  • Motivation: Unlike qPCA, which uses phase estimation and density matrix exponentiation, VQSE requires only one copy of ρ and n qubits per iteration.This design addresses the gate-depth and copy requirements associated with qPCA.
  • Theoretical basis: VQSE minimizes C = Tr(˜ρH), exploiting eigenvalue majorization and Schur concavity so that the transformed state is diagonalized in H's eigenbasis.The Hamiltonian has a non-degenerate spectrum, and minimizing the cost maps the eigenbasis of ρ to that of H.
  • Optimization: The algorithm trains V(θ) in a hybrid quantum-classical loop while adaptively updating H using measurement information to address barren plateaus and local minima.The authors report numerically that adaptive Hamiltonians outperform fixed Hamiltonians in their training experiments.
  • Applications: The introduction motivates VQSE with principal component analysis and numerical applications including error mitigation and entanglement spectroscopy.The error-mitigation application re-purifies a noisy state by targeting its largest-eigenvalue eigenvector.
  • Motivation: VQSE targets the m-largest eigenvalues of an n-qubit density matrix and a circuit that approximately prepares their associated eigenvectors.The target number satisfies m ≪ 2^n.
  • Algorithm output: After optimization, standard-basis measurement estimates eigenvalues from outcome frequencies, while applying V†(θopt) to basis states prepares the corresponding eigenvectors.The procedure reads out the largest measured probabilities as eigenvalue estimates and uses the optimized inverse circuit for eigenvector preparation.

B. The VQSE algorithm

VQSE uses a hybrid quantum-classical loop to optimize a parameterized circuit that approximately diagonalizes a density matrix, then reads out its largest eigenvalues and prepares corresponding eigenvectors.

  • Inputs and outputs: VQSE takes an n-qubit state ρ, a target count m, and an ansatz V(θ), returning estimates of the m-largest eigenvalues and associated eigenvectors.The method assumes m grows at most polynomially with n, although m can theoretically reach 2^n.
  • Hybrid optimization: The optimization evaluates C(t, θ) = Tr[H(t)V(θ)ρV†(θ)] while a classical computer updates θ using the measured cost or its gradient.H(t) is diagonal in the standard basis, efficiently measurable, and has non-degenerate, non-negative lowest m eigenenergies.
  • Optimization termination: Because optimization must terminate, θopt only approximately satisfies the ideal solution condition, but the algorithm provides a verification procedure to quantify solution quality.This approximation arises from imposing a practical termination condition on the optimization loop.
  • Eigenvalue readout: At the optimized parameters, measuring V(θopt)ρV†(θopt) in the standard basis identifies the m largest outcome probabilities as eigenvalue estimates.These probabilities are diagonal elements of the transformed state, not its eigenvalues directly.
  • Statistical estimation: Finite-shot estimation uses outcome frequencies, with smaller inferred eigenvalues producing larger probabilities of a given relative error for fixed Nruns.Increasing Nruns reduces the probability of large relative errors, and a suitable choice can control error below δ for all m eigenvalues.
  • Eigenvector preparation: For each selected bitstring zi, V(θopt)† applied after preparing zi from |0⟩^⊗n produces the corresponding inferred eigenvector.The inferred eigenvalues can be stored classically, whereas extracting information about the quantum-prepared eigenvectors requires measurements.

C. Cost functions

VQSE cost functions balance trainability against solution-space degeneracy: local Hamiltonians avoid barren plateaus but create many local minima, while adaptive Hamiltonians progressively combine local and measurement-updated global terms. The cost also provides an operational route to controlling eigenvalue and eigenvector errors.

  • Fixed Hamiltonians: Global Hamiltonians offer a larger solution space through an m-dimensional non-degenerate sector and a (2^n−m)-fold degenerate level.The arbitrary unitary action on the degenerate subspace creates many optimal unitaries.
  • Fixed Hamiltonians: Fixed local Hamiltonians avoid barren plateaus for shallow hardware-efficient circuits but make optimization harder by reducing solution-space degeneracy.Non-degeneracy helps distinguish more eigenvalues, but leaves fewer valid solutions.
  • Fixed Hamiltonians: Fixed local and global costs can have many local minima because partial eigenvalue assignments produce incomplete solutions, especially for large m.This trainability problem motivates the adaptive cost function.
  • Adaptive Hamiltonian: The adaptive Hamiltonian is updated from measurements during optimization to mitigate barren plateaus, local minima, and the restricted solution space caused by non-degeneracy.Its global component is updated using computational-basis measurement outcomes from the transformed state.
  • Adaptive Hamiltonian: A slowly growing interpolation keeps the early Hamiltonian local and trainable, then adaptively changes the global term so local minima become global minima before ending with H(1)=HG(1).At the endpoint, the spectrum becomes highly degenerate and the solution space increases.
  • Operational meaning: Small VQSE cost values imply small eigenvalue and eigenvector errors through the paper’s upper-bound analysis.The bound uses the lowest m Hamiltonian energies and connects the cost to the inferred eigensystem quality.

D. Verification of solution quality

VQSE solution quality can be verified with an upper bound on eigenvalue and eigenvector errors, using estimates obtained from additional copies of the density matrix and the eigenvalue-readout process.

  • Verification procedure: Verification is optional because it requires 2^n qubits, whereas the main VQSE algorithm requires only n qubits.The verification procedure therefore adds a substantially larger quantum-resource requirement than optimization.
  • Verification procedure: The tighter error bound can be estimated by measuring Tr[ρ^2] once and reusing eigenvalue estimates already produced during VQSE readout.Taking b_m>m requires tracking frequencies for additional bitstrings.

E. Ansatz

VQSE uses state-agnostic hardware-efficient ansätze whose parameter count scales linearly with qubit number and circuit depth, but this ansatz family can become difficult to train as systems grow.

  • Hardware-efficient ansatz: The layered hardware-efficient ansatz uses alternating two-qubit gates on neighboring qubits and requires no prior information about ρ.Its parameter count grows linearly with the number of qubits n and layers L.
  • Hardware-efficient ansatz: Hardware-efficient blocks can use controlled-Z gates with y rotations or CNOT gates with three-rotation single-qubit gates.These are the two parameterizations used in the reported numerics.
  • Trainability: The layered hardware-efficient ansatz can develop trainability problems as system size increases.Suggested mitigations include parameter initialization, layer-by-layer training, correlated parameters, and variable-structure ansätze.
  • Trainability: QAOA is identified as a possible future ansatz because VQSE optimization is an energy-minimization problem, but it was not used in the reported heuristics.The proposed construction uses H(t) as the problem Hamiltonian with a standard mixing Hamiltonian.

F. Optimization

VQSE uses gradient-based optimization and compares fixed and adaptive Hamiltonians across spectrum estimation, error mitigation, and entanglement spectroscopy. The adaptive strategy generally achieves lower errors or better convergence, while the applications demonstrate state purification and detection of quantum critical points.

  • Optimization: Gradient-based optimization is used because gradients for the VQSE cost function can be determined analytically.The authors use gradient-based heuristics rather than gradient-free methods.
  • Quantum principal component analysis: For n = 6, 8, and 10 qubits, adaptive Hamiltonians achieve smaller relative and absolute eigenvalue-estimation errors than fixed Hamiltonians.The adaptive errors are two orders of magnitude smaller for n = 6 and one order smaller for n = 8; for n = 10 they are approximately 10^-5 relative and 10^-7 absolute.
  • Quantum principal component analysis: Adaptive Hamiltonians require fewer runs-per-success for smaller errors and outperform fixed Hamiltonians at sufficiently large inverse error.For up to 8 qubits, approximately 100 repetitions achieve errors of order 10^-6, while the small-error regime can show exponential growth in runs-per-success.
  • Error mitigation: In noisy W-state preparation, VQSE reduces the cost function and produces a re-purified state with higher fidelity to the exact W state than the noisy input.The experiment averages 10 instances and uses an adaptive Hamiltonian updated every 10 iterations.
  • Entanglement spectroscopy: VQSE applied to an XY-model ground state estimates entanglement spectra and detects quantum critical factorization points.The reported factorization points are h/Jx ≈ 0.76 for the ferromagnetic case and h/Jx ≈ 1.43 for the antiferromagnetic case.

III. DISCUSSION

VQSE is presented as a qubit-frugal method for extracting density-matrix eigensystems, with applications to PCA, entanglement spectroscopy, and error mitigation. Numerical results support adaptive-cost optimization, critical-point detection, and W-state repurification, while several extensions remain open.

  • Contributions: VQSE extracts the m largest eigenvalues and associated eigenvectors of ρ using one copy of ρ, requiring only n qubits per iteration.Its cost function derives from diagonalization and majorization, and adaptive updates target barren plateaus and local minima.
  • Applications: For PCA on n = 6, 8, and 10 qubits, adaptive-cost training produced smaller errors than training with a fixed cost function.The experiments estimated the six largest eigenvalues.
  • Applications: VQSE entanglement spectroscopy detected quantum critical points in an XY-model spin chain using eigenvalues of a reduced state.The reduced-state spectrum provides the basis for the entanglement-spectroscopy application.
  • Applications: VQSE repurified a noisy W-state preparation, improving fidelity with the target W state and reducing noise effects.The procedure identifies the circuit preparing the eigenvector with the largest eigenvalue.
  • Scope and future work: VQSE is described as qubit-frugal relative to qPCA, VQSD, and QSVD, but classical-data PCA depends on efficiently preparing the covariance-matrix state.The discussion also identifies combining VQSE with other error-mitigation techniques and testing noise resilience as future work.

A. Operational meaning of the cost function

The cost function is interpreted through majorization and Schur-convexity: estimated eigenvalues and diagonal elements are related by inequalities that connect cost to eigensystem error. These relations yield the stated error bound.

  • Majorization: The eigenvalues of a positive-semidefinite operator majorize its diagonal elements, enabling inequalities between exact eigenvalues and estimated diagonal entries.The argument applies this relation to the leading m-dimensional vectors.
  • Cost-function bound: Schur-convexity of the dot product with an ordered vector converts majorization into a bound involving the VQSE cost function.The cost is expressed as C = Σ_i E_i p_i.
  • Error bound: The ordering of Hamiltonian energies ensures that contributions outside the targeted m eigenvalues enter the cost inequality with the required sign.The derivation assumes E_i ≥ E_{m+1} for i ≥ m + 1.
  • Error bound: Combining the majorization inequalities with bounds on the eigenvalue norm and purity produces the eigenvalue-error bound.The derivation uses λ_m · λ_m ≤ λ · λ = Tr[ρ^2].

B. Verification of solution quality

The verification derives a second bound on eigensystem error by relating eigenvector residuals to estimated eigenvalues and the purity of ρ. A majorization argument then shows this bound is tighter than the earlier one.

  • Error bound: The eigensystem error is bounded by the sum of squared residual norms for the inferred eigenvectors.The residual is defined from the difference between ρ|eλ_i⟩ and eλ_i|eλ_i⟩.
  • Purity relation: Trace invariance gives Σ_i⟨eλ_i|ρ^2|eλ_i⟩ = Tr[ρ^2], linking the residual-based expression to the state purity.The equality follows from evaluating the trace in the inferred eigenvector basis.
  • Majorization: Majorization of a truncated eigenvalue vector by the estimated spectrum yields bλ · bλ ≤ bλ · eλ ≤ eλ · eλ.The dot product with an ordered vector is used as a Schur-convex function.
  • Bound comparison: The resulting bound in Eq. (17) is tighter than the bound in Eq. (16).The comparison follows by showing that the negative term on the right-hand side of Eq. (17) is larger.

C. Gradient of the cost function

The cost gradient is made accessible through parameter shifts, allowing expectation values at shifted circuit parameters to determine each derivative. The derivation applies this rule to layered hardware-efficient circuits.

  • Parameter-shift rule: The derivative with respect to a circuit angle is obtained by shifting that parameter by ±π/2 and measuring the corresponding cost expectation values.This makes gradient values efficiently accessible without explicitly differentiating the full circuit.
  • Ansatz: In the layered hardware-efficient ansatz, V(θ) consists of L layers of two-qubit gates acting on alternating neighboring pairs.Each two-qubit gate is decomposed into gates from a chosen alphabet with continuous parameters.
  • Gradient derivation: Writing the circuit around the differentiated gate as V_L(θ_L)B_μ(θ_μ)V_R(θ_R) isolates the parameter dependence needed for the gradient derivation.The circuit derivative factors through the derivative of the selected gate.
  • Gradient derivation: The final gradient expression is the difference between cost expectation values evaluated at the positive and negative parameter shifts.It is expressed using H(t), V(θ±), ρ, and the corresponding adjoint circuits.

D. Algorithm for the adaptive cost function

The adaptive VQSE algorithm updates the Hamiltonian during optimization using stepwise timing, measured state information, and a classical optimizer. It returns parameters that minimize the cost function after the prescribed iteration loop.

  • Adaptive updates: Every adapting step, it measures V(θ)ρV†(θ) in the standard basis and constructs H_G(t) from the measured basis states and parameters q_i.
  • Inputs: The algorithm takes ρ, a trainable unitary V(θ), timestep and adaptation parameters, a local Hamiltonian H_L, constants {q_i}, and a classical optimizer as inputs.The timestep is δt = 1/Nmax and the adapting stepsize is t_s = 1/s.
  • Optimization loop: At each iteration, the optimizer receives the cost function and current parameters, returns minimizing parameters, and updates θ until the loop reaches t = 1.
  • Initialization: The procedure initializes θ randomly, sets H(t) to H_L, and begins at t = δt.
  • Adaptive updates: The time-dependent Hamiltonian is updated as H(t) ← (1 − t)H_L + tH_G(t), combining the local Hamiltonian with the measured-state-derived Hamiltonian.
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