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Quantum Circuit Learning

Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, Keisuke Fujii

arXiv:1803.00745v3quant-ph

TL;DR

Machine learning on near-term quantum processors requires a framework compatible with low-depth circuits while representing complex nonlinear functions. QCL hybridizes a quantum circuit with classical optimization, and theory plus simulations show that it can approximate nonlinear functions and fit quantum-system dynamics. The framework therefore provides a near-term approach to quantum machine learning.

  • Problem

    Existing quantum machine-learning approaches can require high-depth circuits, motivating a near-term framework for machine learning with low-depth quantum circuits.

  • Method

    QCL encodes data into a quantum circuit, tunes its circuit parameters iteratively with a classical computer, and uses measured observables to produce outputs.

  • Results

    QCL theoretically approximates analytical functions and, in simulations, learns nonlinear functions, performs classification, and fits three observables of a ten-spin system using a six-qubit circuit.

  • Takeaways & Limitations

    QCL uses the exponentially large function space of quantum systems to represent complex functions and offers a framework for near-term quantum machine learning.

Abstract

from arXiv · show

We propose a classical-quantum hybrid algorithm for machine learning on near-term quantum processors, which we call quantum circuit learning. A quantum circuit driven by our framework learns a given task by tuning parameters implemented on it. The iterative optimization of the parameters allows us to circumvent the high-depth circuit. Theoretical investigation shows that a quantum circuit can approximate nonlinear functions, which is further confirmed by numerical simulations. Hybridizing a low-depth quantum circuit and a classical computer for machine learning, the proposed framework paves the way toward applications of near-term quantum devices for quantum machine learning.

I. INTRODUCTION

The paper introduces quantum circuit learning (QCL), a hybrid framework that uses low-depth quantum circuits for machine learning. Iterative parameter tuning enables nonlinear-function approximation and near-term demonstrations.

  • QCL combines a low-depth quantum circuit with iterative tuning of circuit parameters for machine learning.The framework is designed for near-term quantum devices.
  • Gradient-based optimization tunes the circuit parameters similarly to backpropagation in feedforward neural networks.
  • A sufficiently large quantum circuit can theoretically approximate any analytical function.
  • Numerical simulations demonstrate nonlinear-function learning, simple classification, and fitting three-spin dynamics from a fully connected ten-spin system with a six-qubit circuit.
  • The framework is presented as readily realizable on near-term quantum devices.

A. Algorithm

QCL divides learning between a quantum circuit that computes outputs and a classical computer that updates parameters. Its workflow encodes inputs, applies a parameterized circuit, measures observables, minimizes cost, and evaluates on independent data.

  • QCL supports supervised and unsupervised learning by minimizing a cost function through parameter tuning.Supervised learning compares outputs with teacher data, while unsupervised learning uses a teacher-free objective.
  • The quantum circuit computes the output, while a classical computer updates the circuit parameters.
  • Input data are encoded into an initialized quantum state using an input unitary U(x_i).
  • A parameterized unitary U(θ) transforms the input state into an output state.
  • Measured Pauli-operator expectation values are passed through an output function to produce y(x_i, θ).
  • The circuit parameters are iteratively tuned to minimize the cost between teacher data and circuit outputs.
  • Performance is evaluated using a dataset independent from the training data.

B. Relation with existing algortihms

QCL differs from related hybrid quantum methods in what is optimized. Unlike QRC, which tunes linear output weights, QCL tunes the circuit parameters while matching teacher outputs.

  • QCL avoids the high-depth matrix-inversion approach by minimizing regression costs through iterative optimization.
  • QRC passes optimization to a classical computer but tunes a linear weight vector over observables from fixed quantum dynamics.
  • QCL tunes the whole network through circuit parameters, whereas QRC tunes only its linear output weights.
  • QVE/QAOA minimizes a weighted sum of measured expectation values by tuning circuit parameters, representing a special QCL case without an input unitary.

C. Ability to approximate a function

QCL can represent nonlinear functions by encoding inputs into tensor-product quantum states and transforming their observable expectation values. This structure supports polynomial terms, multivariate products, and bounded coefficient norms under unitary constraints.

  • The approximation argument begins with one-dimensional inputs and is stated to generalize straightforwardly to higher-dimensional inputs.
  • QCL outputs are linear combinations of input coefficient functions, with coefficients constrained by the unitarity of the parameterized transformation.
  • A single-qubit-rotation input state contains terms up to order N in x, allowing arbitrary Nth-order polynomials as observable expectation values after a unitary transformation.Terms involving 1 − x^2 can further improve function approximation.
  • Entangling nonlocal operations transfer the highest-order term x^N into a measurable single-qubit observable and create output nonlinearity.
  • For d-dimensional inputs, tensor-product structure provides exponentially many independent coefficient functions and directly represents products such as x1x2.The construction extends to specified higher-order terms for each input dimension.
  • Unitary constraints restrict each coefficient row to unit norm, which the authors state prevents overfitting similarly to regularization in ridge regression.The comparison identifies the matrix elements uij with classical weights w.

D. Possible quantum advantages

QCL may represent functions more complex than a classical counterpart because its quantum circuit can encode relations associated with universal quantum cellular automata. The paper emphasizes that the actual learning costs and advantageous problems remain to be investigated.

  • QCL directly uses an exponential number of functions in the number of qubits, avoiding their explicit classical manipulation and potentially modeling functions intractable for classical computers.The authors present this as a possible quantum advantage over approaches such as QVE or QAOA.
  • A classical neural network learning a quantum circuit’s input-output relation would generally include universal quantum cellular automata, which cannot be achieved with polynomial-size classical resources relative to QCL size.
  • The paper leaves the learning costs and the concrete learning problems that benefit from this potential advantage for further investigation.

E. Optimization procedure

QCL uses gradient-based optimization to tune circuit parameters, adapting the backpropagation idea to expectation values measured from a parameterized quantum circuit. The gradient can be evaluated through shifted circuit rotations and measurements.

  • Gradient-based optimization is preferred as the parameter space grows, paralleling gradient descent and backpropagation in neural networks.This contrasts with gradient-free methods such as Nelder–Mead used in QVE.
  • For a circuit composed of parameterized unitary transformations, the method formulates the gradient of an observable expectation value with respect to a circuit parameter.
  • The commutator relation enables gradient computation on a quantum circuit even though the commutator itself cannot be evaluated directly.
  • Inserting ±π/2 rotations generated by the parameter’s Pauli product and measuring the corresponding expectation values yields the exact gradient.

III. NUMERICAL SIMULATIONS

Numerical simulations evaluate QCL on nonlinear function fitting, classification, and quantum many-body dynamics using a low-depth circuit. The results show successful approximation across these tasks, with weaker fitting for |x| and a 6-qubit circuit fitting selected dynamics of a 10-spin system.

  • Simulation setup: N = 6 and D = 6 define the numerically simulated quantum circuit used across the prototypical machine-learning tasks.The simulations use six qubits and circuit depth six.
  • Function representation: QCL well approximates x^2, e^x, sin x, and |x|, although the |x| fit is relatively poor.The framework exposes higher-order terms for highly nonlinear functions such as sin x and |x|.
  • Classification: The nonlinear classification task uses 200 teacher samples and classifies outputs from two Pauli-Z expectations after softmax transformation.The teacher labels are two-dimensional vectors for classes 0 and 1, and the circuit output is optimized with a cross-entropy cost.
  • Classification: QCL works for the demonstrated nonlinear classification task, using a 0.5 threshold to distinguish the two classes.Values below and above 0.5 correspond to classes 0 and 1, respectively.
  • Quantum advantage context: Compared with a kernel-trick support vector machine, QCL can utilize an exponentially large number of basis functions under certain constraints.The comparison concerns how basis functions are represented in the two approaches.
  • Quantum many-body dynamics: Three observables of a complex 10-spin system are simultaneously well fitted using three observables from a tuned 6-qubit circuit.The teacher data come from dynamics under a fully connected transverse Ising Hamiltonian.

IV. CONCLUSION

The paper presents QCL as a near-term quantum machine-learning framework that uses low-depth circuits to represent, classify, and fit complex functions and quantum-system dynamics.

  • QCL combines injected nonlinear functions with a low-depth quantum circuit to approximate complex nonlinear functions.
  • Numerical results demonstrate QCL’s ability to represent functions, perform classification, and fit a relatively large quantum system.
  • Theoretical analysis indicates that QCL can support high-dimensional regression or classification tasks described as impractical on classical computers.

Appendix: Unitarity avoids overfitting

The appendix compares QCL with classical regression on noisy fitting tasks using identical basis functions. QCL accepts some output error under a unitarity constraint, whereas classical regression overfits with very large weight vectors.

  • QCL and classical regression are compared using the same basis-function set in a three-qubit fitting experiment.The basis set includes polynomial and square-root terms derived from the input encoding.
  • For noisy 0.5 sin x data, QCL accepts some final-output error, while classical regression overfits the training data.The noise has standard deviation 0.05, and the comparison uses QCL in panels (a) and classical regression in panel (b).
  • The classical regression weight vector reaches ∥w∥≈134 for the noisy sin x task and ∥w∥≈15800 for the noisy x2 task.
  • For noisy x2 data, QCL again avoids the overfitting seen in classical regression.The figure compares QCL in panel (c) with classical regression in panel (d).
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