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Exploring entanglement and optimization within the Hamiltonian Variational Ansatz

Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Felipe Carrasquilla, Yong Baek Kim, Henry Yuen

arXiv:2008.02941v2quant-phcond-mat.str-elcs.CC

TL;DR

Problem-specific circuit structure is important for optimizing VQE ansätze without severe barren plateaus. The paper studies HVA through entanglement spectra and gradient statistics, finding structured optimization, over-parameterization effects, and accurate ground-state approximations across several models.

  • Problem

    Randomly initialized, expressive circuit ansätze can exhibit barren plateaus, motivating evidence on whether problem-specific HVA circuits offer more favorable optimization structure.

  • Method

    The paper analyzes HVA entanglement spectra, energy-gradient statistics, initialization strategies, and optimization landscapes in the TFIM and XXZ models, with an additional MHS demonstration.

  • Results

    HVA shows mild or absent barren plateaus, restricted useful state spaces, increasingly trap-free landscapes when over-parameterized, and at-most-polynomial threshold scaling in the studied TFIM and XXZ models.

  • Takeaways & Limitations

    Entanglement properties and initialization can guide effective HVA optimization, including accurate approximations for critical and power-law-entangled ground states.

Abstract

from arXiv · show

Quantum variational algorithms are one of the most promising applications of near-term quantum computers; however, recent studies have demonstrated that unless the variational quantum circuits are configured in a problem-specific manner, optimization of such circuits will most likely fail. In this paper, we focus on a special family of quantum circuits called the Hamiltonian Variational Ansatz (HVA), which takes inspiration from the quantum approximation optimization algorithm and adiabatic quantum computation. Through the study of its entanglement spectrum and energy gradient statistics, we find that HVA exhibits favorable structural properties such as mild or entirely absent barren plateaus and a restricted state space that eases their optimization in comparison to the well-studied "hardware-efficient ansatz." We also numerically observe that the optimization landscape of HVA becomes almost trap free when the ansatz is over-parameterized. We observe a size-dependent "computational phase transition" as the number of layers in the HVA circuit is increased where the optimization crosses over from a hard to an easy region in terms of the quality of the approximations and speed of convergence to a good solution. In contrast with the analogous transitions observed in the learning of random unitaries which occur at a number of layers that grows exponentially with the number of qubits, our Variational Quantum Eigensolver experiments suggest that the threshold to achieve the over-parameterization phenomenon scales at most polynomially in the number of qubits for the transverse field Ising and XXZ models. Lastly, as a demonstration of its entangling power and effectiveness, we show that HVA can find accurate approximations to the ground states of a modified Haldane-Shastry Hamiltonian on a ring, which has long-range interactions and has a power-law entanglement scaling.

I. INTRODUCTION

The paper studies why Hamiltonian Variational Ansatz circuits can be effective for VQE, focusing on their entanglement structure and optimization behavior. It relates structured ansatz spaces and initialization to optimization performance.

  • Motivation: Random circuit ansätze can express many states but suffer from barren plateaus, with exponentially small gradients in flat cost-landscape regions.These observations motivate problem-specific circuit designs.
  • Study design: The study uses entanglement entropy and entanglement spectrum to examine HVA initialization and optimization in the TFIM and XXZ models.Entanglement is treated as a way to characterize the accessible state space and its classical simulability.
  • Study design: HVA can provide a restricted, effective state space, but successful optimization may depend on initialization, particularly for the 1D XXZ model.For the TFIM, ground-state approximations are largely insensitive to initialization, whereas XXZ requires careful initialization.
  • Hamiltonian Variational Ansatz: HVA is inspired by QAOA and adiabatic computation, using multiple Hamiltonian terms rather than only two non-commuting operators.Its circuit structure is model-specific, so its properties can vary across problems.

III. METHODS & MODELS

The TFIM section defines the model-specific HVA circuit and its initialization. A depth p=N/2 circuit is reported to consistently find the ground state across the tested coupling range.

  • Transverse Field Ising Model: The TFIM Hamiltonian is used as a paradigmatic model for quantum magnetism.The model is studied through a one-dimensional periodic chain.
  • Transverse Field Ising Model: The TFIM has a Z2 spin-flip symmetry and exhibits ferromagnetic, paramagnetic, and critical behavior as g varies.For g<1 the system is ferromagnetic, for g>1 paramagnetic, and g=1 is gapless in the thermodynamic limit.
  • TFIM ansatz: The TFIM HVA applies alternating ZZ and RX rotations to an initial |+⟩⊗N state.The initial state is prepared from Hadamard gates, while ZZ gates are two-local and RX gates are single-qubit rotations.
  • TFIM ansatz: A depth-p TFIM HVA has 2p parameters.The circuit illustration uses N=4 and p=1.
  • TFIM results: p=N/2 consistently finds the TFIM ground state for g∈{0.5, 0.52, . . . , 1.5}.The paper notes exact representability at depth p=N/2 for g=0 and numerical evidence for nonzero g.

2. XXZ-model

The XXZ section constructs an HVA from separately parameterized even and odd bonds and interaction components. At criticality, the model has logarithmic entanglement scaling, while depth p=N/2 gives close ground-state approximations across tested anisotropies.

  • XXZ model: The XXZ Hamiltonian combines Hxx, Hyy, and ΔHzz interaction terms, with Δ controlling spin anisotropy.At Δ=1 the model has SU(2) symmetry and is equivalent to the Heisenberg chain.
  • XXZ ansatz: The XXZ HVA decomposes the chain into even and odd non-overlapping bonds and separately parameterizes Hxx, Hyy, and Hzz.Separate bond parameters improve performance for anisotropic systems because one parameter cannot capture Δ≠1 anisotropy.
  • XXZ ansatz: A depth-p XXZ HVA has 4p parameters and starts from the ground state of the even-bond Hamiltonian.For N=4 and p=1, the corresponding circuit includes a Hadamard gate and CNOT operations on even links.
  • XXZ results: p=N/2 is sufficient for a close ground-state approximation for tested Δ∈{0.5, 0.52, . . . , 1.5} with Δ≠1.The Heisenberg-chain case Δ=1 is also reported as accurately solvable with p=N/2.
  • Criticality and entanglement: At the critical point Δ=1, the XXZ system is gapless and its entanglement entropy has a logarithmic correction S∝log N.A matrix product state with bond dimension D satisfies S≤2 log D, implying polynomially growing required bond dimension.

3. Performance Metrics

The paper uses fidelity to compare an optimized VQE state with the exact ground state and treats fidelity above 99.9% as successful ground-state recovery.

  • Fidelity F compares the optimized VQE state with the exact ground state obtained by exact diagonalization.
  • A fidelity above 99.9% is taken as evidence that the ground state has been successfully found.
  • For the studied models, the ground state is always non-degenerate.
  • The infidelity upper bounds the difference between the ground-state and variational expectation values of any observable.

B. Entanglement

The paper characterizes bipartite quantum entanglement through reduced density matrices, von Neumann entropy, and the richer entanglement spectrum of the entanglement Hamiltonian.

  • Bipartite entanglement entropy is the von Neumann entropy of a reduced density matrix obtained by dividing the system into subsystems A and B and tracing out B.
  • For a pure state, a typical bipartition divides an 8-spin ring into two subsystems.
  • The von Neumann entropy can be expressed using the Schmidt decomposition of a bipartite quantum state.
  • The entanglement spectrum contains richer information than entanglement entropy alone and is defined from the eigenvalue spectrum of the entanglement Hamiltonian.
  • For Haar-random quantum states, the entanglement spectrum follows the Marchenko-Pastur distribution, which describes the asymptotic eigenvalue density of Wishart matrices.
  • Page entropy gives the average entanglement entropy of randomly drawn pure states in the full Hilbert space and depends on subsystem dimensions d_A and d_B.

A. The ansatz space through the lens of entanglement spectrum

The entanglement spectrum reveals that HVA ansatz spaces can retain structured entanglement while expressing target ground states, with model-dependent differences between TFIM and XXZ.

  • VQE effectiveness requires an ansatz space containing the ground state and a cost landscape that avoids local minima and reliably reaches it.
  • The study samples 5000 random parameter sets to calculate HVA entanglement spectra for TFIM and XXZ circuits.
  • Even at low circuit depth, both TFIM and XXZ HVA circuits have enough entangling power to express their ground states.
  • For 16-qubit TFIM, the HVA spectrum remains far from Marchenko-Pastur across depths, indicating a restricted entanglement structure.
  • For XXZ, the average spectrum approaches Marchenko-Pastur as depth increases, so its ansatz space is less restricted than TFIM's.
  • XXZ uses more gates and parameters per layer because its richer physics requires a greater variety of accessible states.
  • Identity initialization reaches fidelity above 99.9%, whereas random initialization moves toward a structured local minimum with 70% fidelity.

B. Over-parameterization in HVA

HVA exhibits an over-parameterization threshold beyond which optimization becomes faster and consistently reaches accurate ground-state approximations across random initializations. The threshold appears to scale at most polynomially with system size for TFIM and XXZ models.

  • B. Over-parameterization in HVA: Over-parameterization can make the optimization landscape almost trap free and improve convergence rates after the parameter count crosses a threshold.This phenomenon is discussed as a property of certain non-convex optimization problems and is observed here for HVA.
  • B. Over-parameterization in HVA: For N = 12 qubits, the worst-converging optimization among 100 random initializations is used to assess convergence at each circuit depth.Rapid oscillations in one figure are attributed to Adam-optimizer artifacts and become less severe with increasing depth.
  • B. Over-parameterization in HVA: Once depth reaches a model-dependent threshold, all 100 random starting points converge to accurate solutions, with convergence speed becoming exponentially fast near that threshold.The threshold is not tight: some lower-depth circuits can still reach high-fidelity states.
  • B. Over-parameterization in HVA: The threshold ˜p(N) appears to scale at most polynomially with system size, contrasting with the (2N)^2 parameter count reported for another over-parameterization setting.After sufficient depth, the iteration count saturates at approximately 100 for all system sizes.
  • B. Over-parameterization in HVA: At depth p = 34 for TFIM and p = 52 for XXZ, every starting point requires on the order of 100 iterations to find the ground state.The figure reports decreasing error bars and a critical depth after which all random initializations converge accurately.

C. Ameliorated barren plateaus in HVA

HVA has a more favorable gradient landscape than random quantum circuits: barren plateaus are absent for TFIM and weakened for XXZ. Its entanglement structure helps explain this behavior, and suitable initialization can avoid vanishing gradients.

  • C. Ameliorated barren plateaus in HVA: Random quantum circuits can exhibit barren plateaus where gradients are exponentially close to zero, making local optimization extremely challenging.This behavior is associated with random circuits forming approximate 2-designs and inheriting Haar-like concentration of measure.
  • C. Ameliorated barren plateaus in HVA: HVA has a more favorable optimization landscape than random quantum circuits because its state manifold can have a restricted entanglement structure.For XXZ, the entanglement spectrum alone does not rule out random-circuit-like barren plateaus.
  • C. Ameliorated barren plateaus in HVA: For TFIM, gradient-variance flatness indicates no barren plateau, while XXZ retains exponential decay that is weaker than in random quantum circuits.The gradient-variance behavior is evaluated over 20 random points per system size and depth.
  • C. Ameliorated barren plateaus in HVA: Identity initialization yields constant gradient variance across qubit numbers and enables reliable accurate solutions, indicating that initialization can circumvent vanishing gradients.The paper contrasts this with the milder but still exponential gradient decay observed for XXZ away from that initialization.

D. The entangling power of HVA circuits

HVA’s entangling power grows with circuit depth and is sufficient for ground-state approximation in the studied models. It also reaches high fidelity for a long-range modified Haldane-Shastry Hamiltonian with power-law entanglement scaling.

  • D. The entangling power of HVA circuits: For 1D gapped systems, ground-state entanglement follows an area law, while critical systems exhibit logarithmic entanglement scaling.In one dimension, the boundary area is constant, so the area-law entropy remains constant as system size increases.
  • D. The entangling power of HVA circuits: HVA entangling power depends on circuit depth and is numerically sufficient to express TFIM and XXZ ground states.Entangling power is used to characterize ansatz expressiveness and efficiency.
  • D. The entangling power of HVA circuits: The modified Haldane-Shastry Hamiltonian tests HVA on long-range interactions and a ground state expected to have power-law entanglement scaling.The same HVA used for XXZ can be applied because of the Hamiltonian’s form.
  • D. The entangling power of HVA circuits: > 99.7% fidelity is achieved for the modified Haldane-Shastry ground state with depth p = N for N = 4, 8, 12, 16.For N = 4, the fidelity is close to machine precision and therefore unstable numerically.

V. CONCLUSION

The paper finds favorable optimization properties for HVA, including mild or absent barren plateaus and an over-parameterized regime with improved, eventually trap-free landscapes. It also identifies scope boundaries involving two-dimensional systems and noise in deep circuits.

  • HVA shows evidence of only mild or entirely absent barren plateaus, unlike commonly used random quantum circuits.
  • Over-parameterization increasingly improves HVA’s optimization landscape and eventually makes it trap free above a system-size-dependent threshold.The threshold scales at most polynomially with system size, unlike the exponential scaling observed for learning Haar random unitaries.
  • HVA numerically finds ground-state approximations for the modified Haldane-Shastry Hamiltonian, which has power-law entanglement scaling.
  • Most one-dimensional quantum many-body systems can be simulated efficiently classically, making two-dimensional systems the proposed crucible for HVA.Preliminary rectangular-lattice results indicate that identity initialization remains effective for the XXZ model and TFIM in two dimensions.
  • Greater circuit depth may improve the energy landscape but also increase coherence-time requirements, multiplicative gate errors, and possible noise-induced barren plateaus.Assessing practical hardware usefulness requires analyzing the trade-off between over-parameterization benefits and noise effects.

A. COMPUTATIONAL DETAILS

The computational study uses TensorFlow-based quantum simulation and Adam optimization with explicit stopping and iteration settings. Experiments examine convergence, infidelity, and entanglement dynamics across circuit depths, system sizes, and model parameters.

  • The circuits are simulated with Zyglrox, a TensorFlow-based quantum simulator, while optimization uses the Adam gradient-based optimizer.Adam adapts learning rates separately for each parameter using estimates of the first and second gradient moments.
  • Optimization stops when |E(θt) − E(θt+1)| < 1 × 10^-13, with a maximum of 15000 iterations.
  • The initial Adam learning rate is r = 0.01, within the investigated range 1 × 10^-3 ≤ r ≤ 4 × 10^-2 for TFIM and XXZ optimization.This range was selected as balancing optimization accuracy and convergence speed.
  • Infidelities are evaluated over g and Δ values from 0.5 through 1.5 for TFIM and XXZ, respectively.The corresponding results use identity initialization and a depth p = N/2 circuit, with a dashed cutoff marking 99.9% fidelity.
  • Convergence experiments measure the ratio of random initializations reaching the ground state as a function of circuit depth p and system size N.A run is counted as converged when ϵres ≤ 1e-4.
  • Entanglement-entropy dynamics are tracked at every circuit layer for identity and random-state initializations, including an over-parameterized p = 8 case.The figures compare these dynamics with maximum possible entanglement and Page entropy for final states exceeding 99.9% fidelity.
  • Converged-state entanglement scaling is compared after p/2 and p layers for critical and non-critical TFIM points and the XXZ model.
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