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Hierarchical Learning for Quantum ML: Novel Training Technique for Large-Scale Variational Quantum Circuits

Hrant Gharibyan, Vincent Su, Hayk Tepanyan

arXiv:2311.12929v1quant-phcs.LG

TL;DR

The paper addresses training challenges in variational quantum circuits by introducing hierarchical learning, which expands circuits incrementally across scales. It reports 27-qubit QCBM loading of a multivariate Gaussian and experiments on IBM devices using Fire Opal optimization. The authors also note that full state tomography limits the prospect of exponential speedup for QCBMs.

  • Problem

    Training variational quantum circuits is difficult amid barren-plateau concerns, while distribution-loading methods face gate-cost and scaling challenges.

    Source: §1 · pp. 2–3

  • Method

    Hierarchical learning incrementally expands variational circuits across scales, paired with an adjoint derivative technique for QCBM training.

    Source: §1 · p. 2, §2.2 · pp. 7–8

  • Results

    27-qubit QCBMs with 1000 parameters achieved 4% accuracy in TV distance when approximating multivariate Gaussian distributions.

    Source: §5 · p. 16

  • Takeaways & Limitations

    The authors suggest distribution loading could serve as a component in larger quantum circuits, transferring segmented datasets for subsequent operations.

    Source: §1 · p. 4

    QCBMs may not achieve exponential speedup because they require full state tomography.

    Source: §1 · p. 4

Abstract

from arXiv · show

We present hierarchical learning, a novel variational architecture for efficient training of large-scale variational quantum circuits. We test and benchmark our technique for distribution loading with quantum circuit born machines (QCBMs). With QCBMs, probability distributions are loaded into the squared amplitudes of computational basis vectors represented by bitstrings. Our key insight is to take advantage of the fact that the most significant (qu)bits have a greater effect on the final distribution and can be learned first. One can think of it as a generalization of layerwise learning, where some parameters of the variational circuit are learned first to prevent the phenomena of barren plateaus. We briefly review adjoint methods for computing the gradient, in particular for loss functions that are not expectation values of observables. We first compare the role of connectivity in the variational ansatz for the task of loading a Gaussian distribution on nine qubits, finding that 2D connectivity greatly outperforms qubits arranged on a line. Based on our observations, we then implement this strategy on large-scale numerical experiments with GPUs, training a QCBM to reproduce a 3-dimensional multivariate Gaussian distribution on 27 qubits up to $\sim4\%$ total variation distance. Though barren plateau arguments do not strictly apply here due to the objective function not being tied to an observable, this is to our knowledge the first practical demonstration of variational learning on large numbers of qubits. We also demonstrate hierarchical learning as a resource-efficient way to load distributions for existing quantum hardware (IBM's 7 and 27 qubit devices) in tandem with Fire Opal optimizations.

1 Introduction

The introduction presents hierarchical learning as a way to train increasingly large variational circuits and reports numerical and hardware experiments for QCBM distribution loading. It frames the approach against training challenges in variational circuits and the gate costs of analytic distribution preparation.

1 Introduction

  • Hierarchical learning incrementally expands circuits from a small number of qubits to larger scales, with the stated aim of improving precision and acquiring parameters at finer scales.The authors say it appears to circumvent barren-plateau challenges that often make training with 25 or more qubits difficult.Source: §1 · p. 2
  • QCBMs encode a probability distribution through computational-basis measurement probabilities, with bitstrings mapped to the distribution domain as binary fractions.The variational objective is typically KL divergence between the target distribution and the generated distribution.Source: §1 · p. 3
  • Analytic preparation of normal distributions can require exponentially many gates, whereas variational loading can reduce gate requirements at the cost of optimization simulation.The learned parameters may also be reused as a distribution-loading library.Source: §1 · p. 3
  • The paper uses an adjoint derivative technique for KL-divergence training, which it describes as linear rather than quadratic in the number of circuit parameters.This is presented as a response to the cost of finite-difference methods in large-scale applications.Source: §1 · p. 3
  • Variational-circuit training faces barren-plateau concerns because gradients of expectation-value objectives in sufficiently deep or random circuits can concentrate around zero.The introduction also notes that the QCBM objective is not an observable expectation value and that its barren-plateau results do not strictly apply.Source: §1 · pp. 2–3
  • The authors report numerically loading a 3-dimensional multivariate Gaussian on 27 qubits, their highest numerical experiment for this task, while noting that barren-plateau arguments do not strictly apply.They assume numerical access to the underlying state.Source: §1 · p. 3
  • Fire Opal optimization improved approximation accuracy on IBM Lagos and was particularly beneficial on IBM Algiers circuits with more than 22 two-qubit gates.The Lagos experiments prepared a normal distribution; Algiers circuits targeted a bimodal mixed Gaussian distribution.Source: §1 · p. 4
  • The authors argue that distribution loading could serve as a component in larger circuits, while noting that full state tomography means QCBMs may not achieve exponential speedup.They describe segmented datasets as transferable to a quantum device for subsequent circuit operations and anticipate hierarchical learning may extend beyond QCBMs.Source: §1 · p. 4

2 Methods

The methods train QCBMs hierarchically, first fitting coarse distributions with the most significant qubits and then expanding the circuit, and use adjoint gradients to scale training. They also define a common TV-distance resolution for comparing stages with different numbers of qubits.

2.1 Hierarchical Learning

  • Hierarchical learning trains a small QCBM on a coarse-grained distribution, then uses its parameters to seed a larger circuit focused on finer detail.The approach exploits the greater influence of the most significant qubits for smooth distributions.Source: §2.1 · p. 4, Figure 1 · p. 5
  • New parameters are initialized to 0, and added qubits are initialized in |+⟩ so probability mass splits between neighboring bins.For smooth distributions, this split generally approximates the finer-grained target distribution.Source: §2.1 · p. 5
  • TV_n compares distributions at a shared resolution, adding |+⟩ qubits when the target has more support and marginalizing least-significant qubits when it has less.The paper tracks TV at the largest qubit resolution across training stages to avoid misleading comparisons at coarser resolutions.Source: §2.1 · pp. 5–6
  • A 9-qubit QCBM with 2D connectivity reached TV9 of ∼0.5%, while added 1D layers quickly yielded only small marginal improvements.Figure 3 reports the 2D result at depth 9; the comparison varies circuit depth.Source: Figure 3 · p. 8

2.2 Adjoint Derivative

  • The adjoint method computes gradients for QCBM losses by expressing them as inner products, avoiding finite differences’ quadratic parameter scaling.The forward and backward passes require a number of gate applications linear in the number of parameterized gates; the method is extended to losses that are not observable expectation values.Source: §2.2 · pp. 6–8

3 Distribution Loading

The section examines QCBM distribution loading, finding that 2D connectivity outperforms ring connectivity and that hierarchical learning performs better than learning all parameters of the final circuit at once. It applies the approach to a 27-qubit multivariate Gaussian and uses adjoint gradients to make large simulations more tractable.

3.1 The Role of Connectivity for Small Scale QCBMs

  • Shallow 1D circuits were not expressive enough, and adding layers beyond 4 did not improve TV distance; in contrast, 2D circuits showed no plateau, with TV-distance variance sensitive to initialization as depth increased.Source: §3.1 · p. 9

3.2 Large Scale Multivariate Distribution Loading

  • Hierarchical learning produced the best QCBM overall and on average compared with training the same 24-qubit final circuit from scratch.The hierarchical run expanded in four stages from 9 to 24 active qubits.Source: §3.2 · pp. 10–11, Figure 5 · p. 11
  • A hierarchical QCBM trained in five stages loaded a 3D multivariate Gaussian on 27 qubits; performance variance was relatively small across runs.Each variable used nine qubits on a 3×3 grid, with HEA connectivity similar to the three-set design in Figure 4.Source: §3.2 · p. 11, Figure 6 · p. 12

3.3 Classical Hardware Used in Simulation

  • Adjoint gradients reduced the 30-qubit, 1,000-parameter, 150-epoch training estimate on an A100 from about 730 hours to 7.3 hours.The method scales linearly rather than quadratically with parameter count, although each adjoint circuit evaluation is 10× slower than a vanilla evaluation.Source: §3.3 · p. 13

4 IBM deployment of QCBM models

The experiments compare QCBM distributions executed on IBM Lagos and Algiers in vanilla, Fire Opal-optimized, and ideal-simulator modes. Fire Opal improves accuracy on Lagos, while on Algiers its advantage appears only for circuits with more than 22 CX gates.

4 IBM deployment of QCBM models

  • The comparison spans vanilla IBM, Fire Opal-optimized IBM, and an ideal classical simulator on 7-qubit Lagos and 27-qubit Algiers.The ansatz uses hardware connectivity to construct circuits requiring minimal numbers of two-qubit gates.Source: §4 · p. 13

4.1 7 qubit Experiments: IBM Lagos

  • For the 7-qubit, 24-CX Lagos circuit, the plotted approximation improves from vanilla IBM to Fire Opal-optimized IBM to the ideal simulator.Figure 9 visualizes the 4 most significant qubits.Source: §4.1 · p. 13, Figure 9 · p. 14
  • Fire Opal-optimized runs have a 1.5–8x advantage over vanilla IBM on Lagos, across four circuits preparing the same normal distribution.The comparison uses TV4; the best outcome from five runs was selected for IBM and Fire Opal runs.Source: §4.1 · p. 13, Figure 8 · p. 14

4.2 27 qubit Experiments: IBM Algiers

  • Algiers circuits were trained hierarchically, starting with 4 qubits and adding 4 qubits at each iteration.Figure 10 diagrams the procedure for a 12-qubit, 22-CX circuit using the Algiers connectivity map.Source: §4.2 · p. 14, Figure 10 · p. 15
  • On Algiers, Fire Opal-optimized runs perform better only for circuits with more than 22 CX gates.The circuits prepare the same bimodal mixed Gaussian distribution, and accuracy is measured with TV5.Source: §4.2 · p. 14, Figure 11 · p. 15
  • For the bimodal distribution, TV5 is used instead of TV4 because twice as many buckets are used for visualization and accuracy measurement.Figure 12 shows amplitudes for the 5 most significant qubits across the three device modes.Source: §4.2 · pp. 14, 16, Figure 12 · p. 16

5 Discussion

The discussion reports hierarchical QCBM training with 27 qubits and 1,000 parameters, achieving 4% TV distance on multivariate Gaussian distributions. It presents larger circuits and other variational applications as future directions.

5 Discussion

  • 27-qubit QCBMs with 1,000 parameters achieved 4% accuracy in TV distance for approximating multivariate Gaussian distributions.The authors report that hierarchical learning and adjoint derivatives enabled training on GPUs; the adjoint technique yielded a quadratic speed-up in derivative computations.Source: §5 · p. 16
  • The authors anticipate scaling to larger circuits and more complex distributions, and plan to test the method on financial datasets.They state that their current results indicate no fundamental obstacles to scaling.Source: §5 · p. 16
  • The authors propose exploring hierarchical training for variational quantum eigensolvers by aligning parameter learning with Hamiltonian energy scales.This is a proposed direction, not a demonstrated application in this study.Source: §5 · p. 16

A Visualizing Shallow Circuit Approximation

A hierarchical-learning circuit on the grid connectivity approximates a bimodal Gaussian mixture with TV10 = 0.007. Its shallow implementation uses 10 qubits, 107 single-qubit gates, and 48 CX gates.

A Visualizing Shallow Circuit Approximation

B All-to-All Connectivity and 3D distribution Loading

An all-to-all ansatz approximates an 18-qubit, three-dimensional multivariate normal distribution with TV18 = 0.0285 using two ansatz layers. The authors note that this connectivity requires many two-qubit gates, even for few layers.

B All-to-All Connectivity and 3D distribution Loading

  • The all-to-all ansatz connects every pair of qubits, but requires many two-qubit gates even for few layers.The authors characterize this connectivity as not very practical.Source: B All-to-All Connectivity and 3D… · p. 21
  • TV18 = 0.0285 for an 18-qubit, three-dimensional multivariate normal distribution using two all-to-all ansatz layers.The example uses 6 qubits per dimension and 100 random seeds; it is a different distribution from the one in Figure 7.Source: B All-to-All Connectivity and 3D… · p. 21, Figure 14 · p. 21

C Total Variation Resolution

The section examines how coarse-graining a univariate Gaussian QCBM affects measured total variation distance. It finds that active-qubit distance can diverge from full-distribution distance as qubit count increases.

C Total Variation Resolution

  • For qubit counts below the cutoff, distance on the active qubits separates dramatically from distance on the full distribution.The section uses this univariate Gaussian case to illustrate why TV_n should be considered across qubit sizes.Source: C Total Variation Resolution · p. 22
  • The comparison trains a 24-qubit univariate Gaussian QCBM, discards qubits one at a time, and compares TV_n with TV_24.TV_n is measured on the remaining qubits; TV_24 is measured on the full distribution after discarding qubits.Source: C Total Variation Resolution · p. 22
  • Around 10 qubits, TV_n and TV_24 begin to diverge, and increased resolution beyond 10 qubits does not improve full-distribution TV distance.With relatively few qubits, TV_n can be driven down artificially.Source: Figure 15 · p. 22, C Total Variation Resolution · p. 22

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