Quantum Convolutional Neural Networks

Iris Cong; Soonwon Choi; Mikhail D. Lukin

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Quantum Convolutional Neural Networks

Iris Cong, Soonwon Choi, Mikhail D. Lukin

arXiv:1810.03787v2Oct 9, 2018quant-phcond-mat.str-el

TL;DR

The paper develops QCNNs as quantum convolutional architectures for recognizing quantum phases and optimizing quantum error correction. QCNNs use convolution, pooling, and variational optimization, accurately identify a phase transition, and learn error-correction codes that outperform known codes for some error models.

Problem
The paper asks whether a quantum circuit can detect transitions between quantum phases and optimize error correction for specific error models.
Method
QCNNs combine quasi-local translationally invariant unitaries, measurement-based pooling, and a final fully connected unitary, with parameters optimized by gradient descent; they can jointly optimize encoding and decoding.
Results
QCNNs accurately identify the Haldane-to-paramagnetic phase transition and significantly outperform known codes for anisotropic correlated errors, while matching or improving on baselines in other tested models.
Takeaways & Limitations
QCNNs provide a framework for quantum phase recognition and error-correction design, with constructions extending to phases whose fixed-point states have suitable tensor-network representations.
Takeaways & Limitations
The reported error-correction demonstrations assume independent single-qubit errors, with one model additionally including correlated two-qubit errors.

Abstract

from arXiv · show

We introduce and analyze a novel quantum machine learning model motivated by convolutional neural networks. Our quantum convolutional neural network (QCNN) makes use of only $O(\log(N))$ variational parameters for input sizes of $N$ qubits, allowing for its efficient training and implementation on realistic, near-term quantum devices. The QCNN architecture combines the multi-scale entanglement renormalization ansatz and quantum error correction. We explicitly illustrate its potential with two examples. First, QCNN is used to accurately recognize quantum states associated with 1D symmetry-protected topological phases. We numerically demonstrate that a QCNN trained on a small set of exactly solvable points can reproduce the phase diagram over the entire parameter regime and also provide an exact, analytical QCNN solution. As a second application, we utilize QCNNs to devise a quantum error correction scheme optimized for a given error model. We provide a generic framework to simultaneously optimize both encoding and decoding procedures and find that the resultant scheme significantly outperforms known quantum codes of comparable complexity. Finally, potential experimental realization and generalizations of QCNNs are discussed.

QCNN CIRCUIT MODEL

QCNN extends CNN principles to quantum circuits through translationally invariant quasi-local convolutions, measurement-based pooling, hierarchical size reduction, and a final fully connected measurement. Its architecture uses O(log(N)) parameters and connects naturally to MERA and nested quantum error correction.

QCNN architecture: QCNN applies finite-depth translationally invariant quasi-local unitaries, measures some qubits for pooling, and uses outcomes to control rotations on nearby qubits.Convolution and pooling repeat until few qubits remain, followed by a fully connected unitary and measurement of fixed output qubits.
Parameter efficiency: A QCNN classifier for N-qubit inputs uses O(log(N)) parameters, enabling efficient learning and implementation.The circuit structures remain fixed while the unitaries are learned.
Training: Learning minimizes mean-squared error between binary labels and expected QCNN outputs by successively optimizing the circuit unitaries.The optimization can use gradient descent after random initialization.
MERA and QEC: QCNN pooling measurements act as quantum-error-correction syndrome measurements that determine correction unitaries on remaining qubits.This gives the circuit a combined interpretation as a MERA variational ansatz and nested QEC procedure.

DETECTING A 1D SPT PHASE

The paper applies an exact QCNN to recognize a one-dimensional Z2 × Z2 symmetry-protected topological phase. Compared with string-order measurements, the QCNN requires fewer input copies, especially near the phase transition, with improvements associated with iterative error correction.

Exact circuit: An exact QCNN recognizes the Z2 × Z2 SPT phase using controlled-phase gates, X-basis Toffoli gates, measurement-based pooling, and a fully connected layer.The circuit measures the local structure Zi−1XiZi+1 in its final layer.
Task: The QPR task asks whether an unknown ground state from the phase diagram belongs to the target SPT phase.At h1 = h2 = 0, the ground state is the 1D cluster state and the Hamiltonian terms commute.
Sample complexity: QCNN requires substantially fewer input copies than string-order parameters throughout the parameter regime, especially near criticality.String-order expectations vanish near the phase boundary, where projection noise increases the repetitions needed for detection.
Sample complexity: QCNN sample complexity improves with depth and can decrease relative to string-order measurements by a factor scaling exponentially with depth in accessible regimes.The improvement is attributed to iterative QEC at each depth and is limited by finite-size effects in the simulations.
Mechanism: The QCNN measures a multiscale string-order parameter formed from products of exponentially many string-order parameters that remains sharp to the phase boundary.This distinguishes its depth-dependent behavior from measurements of simple observables.

MERA and QEC

The QCNN combines MERA-like multiscale state reduction with nested quantum error correction, using measurements to detect errors and conditional unitaries to correct them. This structure supports both analytical classification behavior and learned phase-recognition applications.

MERA: The QCNN contains the MERA representation of the cluster state as a stable fixed point under repeated convolution-pooling layers.Each layer reduces system size while preserving the cluster state on unmeasured qubits and producing deterministic measurement outcomes.
QEC: A single X error is identified by the first pooling layer and corrected through controlled unitaries propagated through the circuit.Multiple sufficiently separated errors can also have reduced density after repeated convolution and pooling.
Training: The learned demonstration used 40 solvable training points for an N = 15, depth d = 1 QCNN with 1309 trainable parameters.The parameters were initialized randomly and optimized with gradient descent.
Training: The trained QCNN reproduced the 2D phase diagram across the full parameter regime despite training only on solvable points that did not cross the lower phase boundary.The authors attribute this example to the architecture’s exponentially reduced parameter count and its ability to avoid overfitting.

OPTIMIZING QUANTUM ERROR CORRECTION

The paper treats a QCNN and its inverse as decoder and encoder channels, then optimizes both against a specified noise process. The learned code matches known codes for symmetric errors and improves performance for anisotropic and correlated error models.

Framework: A QCNN and its inverse define decoding and encoding channels between physical qubits and a logical qubit.Encoding adds qubits in predetermined states, while decoding performs measurements.
Framework: The variational procedure simultaneously optimizes encoding and decoding and can be evaluated experimentally without knowing the noise channel.
Evaluation: A two-layer QCNN with N = 9 physical qubits and 126 variational parameters was compared with repetition and Shor codes.The tested models included symmetric, anisotropic, and correlated single- and two-qubit errors.
Results: For correlated errors, the optimized QEC code significantly outperformed known codes, whereas the Shor code performed worse than no error correction.
Implementation: QCNNs require efficient many-body-state preparation, multiscale two-qubit gates, and projective measurements, all demonstrated on programmable platforms with N ≥50 qubits.
Implementation: A conservative neutral-atom estimate indicates that a d = 4 QCNN on N ∼100 qubits is feasible within the available multi-qubit-operation budget.

OUTLOOK

The authors identify extensions of QCNNs to higher-dimensional phases, more exotic topological systems, and alternative architectures. They also propose relaxing translation invariance, adding ancillas, optimizing fault-tolerant operations, and improving gradient estimation.

Generalizations: QCNNs could be generalized to higher dimensions to support phases with intrinsic topological order such as the toric code.
Generalizations: The framework could target nonlocal order parameters for lesser-understood phases including quantum spin liquids and anyonic chains.
Architectures: Relaxing translation invariance would increase the parameter count to O(N), while ancilla qubits could implement parallel feature maps.
Future directions: Future extensions include optimizing fault-tolerant operations on QEC code spaces and adopting more efficient gradient schemes than finite differences.

Phase Diagram and QCNN Circuit Simulations

The phase-recognition simulations use DMRG and matrix-product-state calculations to evaluate QCNN outputs for Haldane-chain systems. An exact circuit construction is obtained by mapping spin-1 degrees of freedom into a symmetry-compatible spin-1/2 representation.

Phase diagram: Infinite-size DMRG identifies phase boundaries from sharp peaks in the second derivative of ground-state energy density.The simulations use maximum bond dimension 150.
Simulation: Figure 7 evaluates exact QCNN outputs at depths d = 1, ..., 4 for N = 54 spin-1 particles.
Phase diagram: For the Haldane chain, small ω/J gives the SPT phase, whereas sufficiently large ω/J produces a paramagnetic ground state.
Circuit construction: The spin-1/2 mapping uses a quasi-local isometry that converts the distinct symmetry representations into the cluster-model representation.It satisfies URxU† = Xodd and URyU† = Xeven.
Circuit construction: The Haldane-chain QCNN applies the isometry U before the main-text QCNN circuit.
Simulation: The simulation identifies the critical point as ω/J = 1.035 ± 0.005 using matrix-product-state calculations with bond dimension D = 160.The QCNN is reported to provide accurate phase-transition identification.

Multiscale String Order Parameters

The QCNN transforms measured Pauli operators recursively in the Heisenberg picture, producing a multiscale string order parameter composed of exponentially many string-operator products.

The QCNN postpones pooling measurements and replaces them with controlled unitary gates, making the circuit equivalent to measuring a non-local observable.
Recursive conjugation through convolution-pooling units transforms a string operator into weighted products of operators at the preceding depth.
The measured operator is a sum of exponentially many string-operator products whose coefficients are determined recursively.
The resulting observable is a multiscale string order parameter that can yield sharp classification even when the correlation length reaches 3^d.
The illustrated QCNN parameterization uses four convolution layers, a pooling layer, and a fully connected layer.

Demonstration of Learning Procedure for QPR

The QPR learning procedure uses a hyperparameterized convolution-pooling architecture, finite-difference gradients, and iterative gradient descent to optimize the circuit parameters.

Each convolution-pooling layer reduces system size by a factor n, with the n = 3 example repeated for QCNN depth d.The fully connected layer acts on the remaining N/n^d qubits.
The pooling layer measures n − 1 qubits in each block of n and applies outcome-dependent unitaries to the remaining qubit.
The final classification output is obtained by measuring a fixed qubit after an arbitrary fully connected unitary on the reduced system.
The simulations use N = 15 spins and depth d = 1 because classical simulation of the circuits is resource-intensive.
Parameters are initialized randomly, gradients are estimated by finite differences, and gradient descent continues until successive errors differ by about 10^-5.The finite-difference computation uses ϵ = 10^-4, with an adaptive learning rate.

Construction of QCNN Circuit

The QCNN circuit is constructed by enforcing fixed-point renormalization and quantum-error-correction criteria, using the cluster-state MERA structure and stabilizer-code interpretation.

The fixed-point criterion requires a cluster state on L spins to map to a cluster state on L/3 spins with deterministic measurement outcomes.
The QEC criterion requires single-site symmetry-commuting errors to leave a smaller cluster state while producing at least one |1⟩ measurement outcome.
These fixed-point and QEC properties are identified as desirable for quantum-circuit implementations of renormalization-group flow in QPR.
The cluster-state ground-state manifold is treated as a stabilizer-code space, and the circuit is specified to detect and correct single-qubit X errors.
For translationally invariant isometric matrix product states, the MERA representation and its isometry can be explicitly constructed, with 3-to-1 maps generalizing to arbitrary n-to-1 isometries.

QCNN for General QPR Problems

The MERA-and-QEC interpretation extends QCNN construction beyond the cluster-state example to phases with suitable fixed-point tensor-network representations and local-error correction schemes.

For phases whose RG fixed-point wavefunction has an isometric or G-isometric tensor-network representation, a corresponding QCNN can be systematically constructed.The stated family includes all 1D SPT and 2D string-net phases.
The generic construction combines the MERA for the fixed-point wavefunction with nested QEC based on measurements of parent-Hamiltonian terms.
Input states with error density below the QEC threshold flow toward the renormalization-group fixed point, and the resulting QCNN can be optimized through the learning procedure.
Initializing with a fixed-point MERA and one nested-QEC choice may make learning serve to optimize the QEC scheme toward the target phase boundary.

Experimental Resource Analysis

The QCNN circuit can be implemented with Rydberg-mediated gates, and its resource requirements are explicitly counted for an N-spin input.

Postponing pooling measurements to the circuit end is used when analyzing the gate depth of the cluster-model QCNN in a Rydberg-atom implementation.
Rydberg blockade-mediated controlled gates implement the required multi-qubit QCNN operations through native gates and single-qubit conjugations.The construction uses CzZij, CzCzZijk, and rotation-conjugated CxZij and CxCxXijk gates.
For an input of N spins, the kth convolution-pooling unit applies 4N/3^(k−1) CzZij gates, N/3^(k−1) CxCxXijk gates, and 2N/3^(k−1) CxZij layers.
The implementation needs no SWAP gates because the Rydberg interaction is long-range.

Demonstration of Learning Procedure for QEC

The QEC demonstration trains a compact two-layer QCNN by optimizing its layers sequentially under several error models, including anisotropic and correlated noise. The optimized code matches known codes for isotropic errors and improves on them for anisotropic settings, especially with correlations.

The first convolution and fully connected unitaries are generic three-qubit operations, each parameterized by 63 Gell-Mann coefficients.
The study evaluates equal-probability single-qubit errors, anisotropic single-qubit errors, and anisotropic errors with additional local two-qubit correlations.
The QCNN code is trained across anisotropic models with fixed total input error p_tot = 0.001 while varying the ratio p_x/p_tot and comparing logical error rates with the Shor code.
For strongly anisotropic models, QCNN outperforms the Shor code; for nearly isotropic models, the Shor code is optimal and QCNN reaches the same logical error rate.
The optimization proceeds sequentially: U1 is chosen to make the effective channel M_U1 approximately one-axis dominated, then U2 is optimized to maximize logical-state fidelity.