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Capacity and quantum geometry of parametrized quantum circuits

Tobias Haug, Kishor Bharti, M. S. Kim

arXiv:2102.01659v2quant-phcs.LGstat.ML

TL;DR

Variational quantum algorithms on NISQ devices need circuits whose capacity and trainability can be assessed, but tools for evaluating initialization strategies are lacking. This paper uses quantum-geometric measures to compare parametrized circuits and reports gate-dependent expressibility, gradient behavior, and a way to prune redundant parameters.

  • Problem

    Tools for evaluating the capacity of parametrized circuits and initialization strategies for NISQ variational algorithms are lacking amid challenges including barren plateaus.

    Source: §I · p. 1

  • Method

    The paper derives global parameter dimension and local effective quantum dimension from the quantum Fisher information and applies them to hardware-efficient circuits and parameter pruning.

    Source: §I · p. 1, §VI · p. 8

  • Results

    Circuit structure and entangling gates affect expressibility; tuning parameters from zero toward random values reveals a range with large gradients and nearly maximal effective quantum dimension, while deep-circuit QFI transitions accompany QNG decay.

    Source: §I · p. 1, §VI · pp. 7–8

  • Takeaways & Limitations

    The pruning algorithm reduces PQC parameter count while preserving parameter dimension, reducing gradient and QFI computation and circuit depth.

    Source: §VI · p. 8

    The reported parameter-tuning study considered θ = 0; the authors note that correlated-parameter PQCs could show similar behavior.

    Source: §VI · p. 7

Abstract

from arXiv · show

To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension, which reveals the expressive power of circuits in general as well as of particular initialization strategies. We assess the expressive power of various popular circuit types and find striking differences depending on the type of entangling gates used. Particular circuits are characterized by scaling laws in their expressiveness. We identify a transition in the quantum geometry of the parameter space, which leads to a decay of the quantum natural gradient for deep circuits. For shallow circuits, the quantum natural gradient can be orders of magnitude larger in value compared to the regular gradient; however, both of them can suffer from vanishing gradients. By tuning a fixed set of circuit parameters to randomized ones, we find a region where the circuit is expressive, but does not suffer from barren plateaus, hinting at a good way to initialize circuits. We show an algorithm that prunes redundant parameters of a circuit without affecting its effective dimension. Our results enhance the understanding of parametrized quantum circuits and can be immediately applied to improve variational quantum algorithms.

I. INTRODUCTION

The paper introduces parameter dimension and effective quantum dimension as measures of circuit capacity, derived from the quantum Fisher information. Applying these measures to hardware-efficient circuits, it reports differences by circuit structure and entangling gate, a QFI transition affecting the quantum natural gradient, and a parameter-pruning algorithm.

I. INTRODUCTION

  • The motivating setting is variational quantum algorithms on NISQ devices, where optimization faces barren plateaus and the classical optimization component has been shown to be NP-hard.The introduction notes that tools for evaluating initialization and layer-wise learning strategies are lacking.Source: §I · p. 1
  • The paper uses parameter dimension as a global capacity measure and effective quantum dimension as a local measure of accessible states, deriving both from the quantum Fisher information metric.The metric also provides a route to the quantum natural gradient for optimization.Source: §I · p. 1
  • Circuit structure and entangling-gate choice produce marked differences in expressive power and in how well particular initialization strategies perform.The investigated circuits also differ in redundant-parameter counts and convergence toward random states as layer count increases.Source: §I · p. 1
  • Deep circuits exhibit a QFI-spectrum transition associated with decay of the quantum natural gradient; for shallow circuits, the QNG can greatly exceed the regular gradient, though both may vanish.The reported transition occurs when the parameter dimension reaches its maximum possible value.Source: §I · pp. 1–2
  • The investigated hardware-efficient circuits consist of layers of parametrized single-qubit rotations followed by two-qubit entangling gates.The considered entanglers include CNOT, CPHASE, and iSWAP, arranged in different geometries.Source: §I · p. 2
  • Tuning circuit parameters from zero toward random values identifies a range combining large gradients with nearly maximal effective quantum dimension, while pruning can reduce circuit parameters and depth.The pruning algorithm preserves parameter dimension; reduced depth can ease noise impact.Source: §I · p. 2

II. PARAMETRIZED QUANTUM CIRCUITS

A parametrized quantum circuit maps parameters and an initial product state to an N-qubit quantum state. Its expressibility can be assessed by comparing its state distribution with the Haar-random distribution, while gradient variance provides context for trainability.

II. PARAMETRIZED QUANTUM CIRCUITS

  • A PQC generates an N-qubit state by applying a unitary U(θ) to the product state |0⟩⊗N.The circuit structure influences which quantum states it can express.Source: §II · p. 2
  • Expressibility can be measured by the distance between the circuit's state distribution and the Haar-random distribution, indicating how well it covers the Hilbert space.For local cost functions, low gradient variance correlates with high expressibility in most cases.Source: §II · p. 2
  • For randomly parameterized PQCs, gradient variance can decrease exponentially with qubit count and decline with layer count until a threshold, after which it remains constant.The passage connects vanishing gradients with expressibility measures, while noting the local-cost-function qualification for the stated correlation.Source: §II · p. 2

III. PARAMETER DIMENSION

Parameter dimension counts the independent parameters a PQC can express in quantum-state space, distinguishing them from parameters that leave the state unchanged. The dimension is bounded by that of a generic N-qubit state, and redundancy measures the dependent fraction.

III. PARAMETER DIMENSION

  • For a generic complex N-qubit state, the parameter dimension is D_C = 2^(N+1) − 2, giving the upper bound D_C ≤ 2^(N+1) − 2 for a PQC.Two of the 2^(N+1) state parameters are dependent because they encode norm and global phase.Source: §III · p. 3
  • Parameter dimension D_C is the number of independent parameters a PQC can express in quantum-state space.For a generic real-valued quantum state, D_C = 2^N − 1.Source: §III · p. 3
  • Redundancy is the fraction of dependent circuit parameters that do not contribute to changing the quantum state.It distinguishes the total parameter count from the state-changing degrees of freedom.Source: §III · p. 3

IV. EFFECTIVE QUANTUM DIMENSION

The effective quantum dimension is a local, QFI-based measure of how many independent state-space directions infinitesimal parameter updates can reach, and it can be lower than the parameter dimension.

IV. EFFECTIVE QUANTUM DIMENSION

  • Along an eigenvector α(i) of F, the state-change rate scales with λ(i); zero eigenvalues leave the state unchanged, while small positive ones are near-singular.The paper associates near-singularities with plateaus where training slows down.Source: §IV · p. 4
  • G_C is the rank of the QFI F(θ), counting the independent state-space directions accessible through infinitesimal parameter updates.For the described layer-wise PQCs, G_C ≤ D_C ≤ M.Source: §IV · p. 4
  • For a single-qubit state, QFI eigenvalues 1 and sin²(θ) give G_C=2 except at θ=nπ, where the vanished ϕ direction leaves G_C=1.At those singular settings, changing ϕ does not change the state.Source: Figure 2 · p. 3, §IV · p. 4
  • In the single-qubit circuit example, the QFI has one nonzero eigenvalue and D_C=1, leaving M−1 parameter directions redundant.Those directions do not change the quantum state.Source: §IV · p. 4
  • For the described PQC types, random parameters provide numeric evidence that G_C≈D_C, allowing D_C to be calculated from G_C at random initializations.Special parameter settings can instead have G_C<D_C.Source: §IV · p. 4
  • Equation (13) is not valid for arbitrary circuits: a non-2π-periodic one-parameter qubit evolution can have D_C=2 but G_C=1.The example's evolution covers all possible single-qubit states despite having only one parameter.Source: §IV · p. 4

V. RESULTS

The results show that circuit depth, entangling-gate choice, and parameter initialization shape expressibility and gradient behavior. They also demonstrate that redundant parameters can be pruned while preserving parameter dimension.

V. RESULTS

  • For chain CNOT circuits, parameter dimension grows linearly with layer count until reaching its maximum at a characteristic p_c, where the QFI spectrum undergoes a transition.Near p_c, the QFI eigenvalue distribution broadens and develops a tail of small eigenvalues; above the transition, those small eigenvalues disappear.Source: §V · p. 5
  • For H = σz 1σz 2, QNG variance remains above regular-gradient variance but decays for p > p_c; both variances vanish exponentially as qubit count increases.The regular-gradient variance reaches a minimum around p ≈ 20 and then remains constant.Source: §V · p. 5
  • Across tested circuit structures, gradient variance decreases with depth to the same constant level, but CPHASE circuits require the most layers to converge, followed by iSWAP and CNOT.The tested structures also share a qualitative QFI transition and an exponential decrease in gradient variance with increasing qubit count.Source: §V · p. 5, Figure 4 · p. 6
  • Redundancy quickly plateaus: iSWAP is consistently low, CPHASE is higher because z rotations commute through its entangling layer, and large-p redundancy helps estimate p_c.For CNOT, redundancy depends strongly on entangling-gate arrangement; the estimate uses large-p redundancy R_C and parameterized rotations per layer b.Source: §V · p. 5
  • With y rotations in a chain, maximal parameter dimension scales exponentially for y CNOT and y iSWAP, but approximately quadratically for y CPHASE.The reported scalings are D_C ∝ 2^N for y CNOT, D_C ∝ 2^(N+1) for y iSWAP, and D_C ∝ N^2 for y CPHASE.Source: §V · p. 6, Figure 5 · p. 6
  • Tuning parameters from zero toward random values makes G_C rise to D_C; near log10(a) ≈ −2.5, G_C is nearly maximal while gradient variance remains large.CNOT and iSWAP reach higher G_C faster with a than CPHASE.Source: §V · p. 6, Figure 6 · p. 6
  • At θ = 0, G_C scales linearly with N for CPHASE, quadratically for CNOT, and at higher polynomial order or even exponentially for iSWAP.The comparison uses nearest-neighbor chain circuits with the number of layers chosen to maximize G_C(θ = 0).Source: §V · pp. 6–7, Figure 7 · p. 7
  • The pruning algorithm removes parameters associated with zero-eigenvalue QFI directions and yields a circuit with as many parameters as the original circuit’s parameter dimension, without changing that dimension.The CPHASE-CHAIN demonstration reports a substantial parameter reduction while preserving D_C.Source: §V · pp. 6–7

VI. DISCUSSION

The discussion uses quantum geometry to compare circuit capacity and trainability, identifying gate-dependent expressiveness, a QFI transition in deep circuits, and parameter pruning that preserves dimension. It also outlines implications and extensions of these findings.

VI. DISCUSSION

  • iSWAP circuits have lower gradient variance than CPHASE circuits, while the same exponential gradient-variance decay extends to Hamiltonians with polynomially many Pauli terms.The authors attribute the gate-dependent expressibility differences to limited single-qubit rotations and demonstrate the broader decay for a many-body Hamiltonian.Source: §VI · p. 7
  • For y-rotation circuits with CPHASE gates, DC scales quadratically with qubit number; parameter redundancy also varies strongly with gate type and arrangement.The authors note that this scaling may imply efficient classical simulation for that circuit type.Source: §VI · p. 7
  • Tuning parameters from zero toward random values yields a range with large gradients and nearly maximal GC, a potential starting point for gradient-based optimization.The reported crossover runs from large gradients and small GC to vanishing gradients and large GC.Source: §VI · p. 7
  • For p > pc, the QFI transition leads to QNG decay; for p < pc, QNG can exceed the regular gradient by orders of magnitude, but both gradients decrease exponentially with qubit number.The transition coincides with disappearance of small QFI eigenvalues and a peak in the variance of their logarithms.Source: §VI · pp. 7–8
  • The authors note that QFI-based updates in imaginary-time evolution and variational quantum simulation may benefit from insights into GC and circuit geometry.They also point to adaptive ansätze as a possible application.Source: §VI · p. 8
  • The proposed algorithm reduces PQC parameter count and depth while keeping DC constant, reducing the QFI size and the number of parameterized gates needed.Removing one parameter reduces the number of QFI elements to measure by M.Source: §VI · p. 8
  • QFI eigenvalue spectra vary with gate type and arrangement; the discussion raises their possible relevance to trainability and generalization, and suggests evaluating other circuit families.The authors identify noisy and convolutional PQCs, optimal control, quantum metrology, and programmable analog simulators as possible extensions.Source: §VI · p. 8

Appendix A: Variational quantum eigensolver

  • VQE seeks a Hamiltonian ground state by minimizing a PQC energy objective, which the Ritz variational principle bounds below by the true ground-state energy.A classical optimizer selects parameters while the quantum device evaluates the energy.Source: Appendix A · p. 10

Appendix B: Quantum Fisher information metric

This appendix describes parameter updates for variational circuits and motivates the quantum natural gradient as a geometry-aware alternative to ordinary gradient descent. Its update uses the QFI metric, which is related to changes in the quantum state.

Appendix B: Quantum Fisher information metric

  • Ordinary gradient descent updates PQC parameters iteratively using the objective's gradient, with a classical optimizer choosing successive parameters from quantum-computed objective values.The update assumes a flat Euclidean parameter space.Source: Appendix B · p. 10
  • The quantum natural gradient uses information about how quickly the quantum state changes with a parameter direction to minimize the objective.It is introduced as an alternative because the parameter space need not be flat Euclidean space.Source: Appendix B · p. 10
  • F(θ) is the Fubini–Study metric tensor, also called the quantum Fisher information metric, and is related to the Fubini–Study distance between pure states.The state derivative |∂iψ⟩ is defined as the partial derivative of |ψ(θ)⟩ with respect to θ_i.Source: Appendix B · p. 10

Appendix C: Further data on the PQCs

Further circuit examples show that rotation axes can affect redundancy and gradient decay, while another circuit pair exhibits matching gradient-variance decay despite different rotation counts. The appendix also reports additional signatures of the QFI transition.

Appendix C: Further data on the PQCs

  • rand(xyw) CPHASE has lower parameter redundancy than rand(xyz) CPHASE because noncommuting (x + y)/2 rotations replace z rotations that commute with CPHASE.The substitution also leads to a faster decrease in gradient variance.Source: Appendix C · p. 10
  • zxz CNOT and rand(xyz) CNOT show the same gradient-variance decay with layer count, although zxz uses three rotations per qubit per layer.The zxz circuit is identified as a common type first introduced in.Source: Appendix C · p. 10
  • Further examples show the QFI transition through a peak in the variance of log eigenvalues and a decay in QNG.Source: Appendix C · p. 10

Appendix D: Variance of gradient of Hamiltonians

The appendix compares gradient variance across Hamiltonians and circuit types. After normalization by the number of Hamiltonian terms, the two Hamiltonian cases have nearly the same variance.

Appendix D: Variance of gradient of Hamiltonians

  • Gradient variance has the same exponential decay for Hamiltonians comprising a polynomial number of Pauli operators.The paper demonstrates this for a many-body Hamiltonian in Appendix D.Source: Appendix D · p. 10
  • After division by the number of terms, gradient variance is nearly the same for the two-qubit and transverse Ising Hamiltonians.Figure 9 plots gradient variance against layer count, using solid curves for the two-qubit Hamiltonian and dashed curves for transverse Ising, for chain-entangled circuits with N = 10.Source: Appendix D · p. 11, Figure 9 · p. 11

Appendix E: Histograms of eigenvalues

The eigenvalue distributions differ by PQC type, with CPHASE showing more pronounced tails across the plotted cases.

Appendix E: Histograms of eigenvalues

  • CPHASE shows more pronounced eigenvalue-distribution tails across the circuit arrangements.The plotted PQCs use nearest-neighbor chain, all-to-all, and alternating nearest-neighbor entangling-gate arrangements.Source: Appendix E · p. 11

Appendix F: Pruning PQCs of redundant parameters

The pruning algorithm removes redundant parameters while preserving the circuit’s parameter dimension and thus its expressiveness.

Appendix F: Pruning PQCs of redundant parameters

  • Applying Algorithm 1 to the CPHASE CHAIN PQC removes redundant parameters and substantially reduces the number of unitaries, while its parameter dimension remains 126 before and after pruning.Source: Appendix F · p. 11

Appendix G: Measuring the quantum Fisher information metric

The appendix reviews methods for measuring the quantum Fisher information (QFI) of parametrized circuits, including Hadamard-test overlaps and a fidelity-based shift rule. It describes their measurement requirements and notes that approximations can reduce the cost in commuting circuits.

Appendix G: Measuring the quantum Fisher information metric

  • The QFI is a positive-semidefinite M × M metric, and the appendix reviews approaches for calculating it.These include a Hadamard test for derivative-state overlaps and a shift-rule approach based on the fidelity Hessian.Source: Appendix G · pp. 11–12
  • The shift-rule method computes the QFI as a fidelity Hessian, using the inversion test to measure state fidelity.The full QFI requires 2M(M − 1) + M fidelity measurements via the shift rule.Source: Appendix G · pp. 11–12
  • Commuting unitaries enable approximate QFI calculations whose diagonal and block-diagonal entries scale linearly with the number of parameters M.The appendix also notes that direct measurements can replace controlled unitaries when avoiding an ancilla is desired.Source: Appendix G · pp. 12–13
  • The Hadamard test evaluates real derivative-state overlaps using an ancilla and controlled unitaries, with the circuit illustrated in Fig. 12.For M parameters, this method requires M(M − 1)/2 such tests, plus M Pauli-string measurements for the other QFI terms.Source: Appendix G · p. 13, Figure 12 · p. 13
  • For the circuit ansatz, diagonal derivative-state overlaps are trivial, while off-diagonal overlaps can be complex and require the Hadamard test.The parameterized rotations use Pauli operators; their relevant overlap terms can also be measured by sampling in a Pauli-rotated computational basis.Source: Appendix G · pp. 12–13

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