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Realizing Quantum Convolutional Neural Networks on a Superconducting Quantum Processor to Recognize Quantum Phases
Johannes Herrmann, Sergi Masot Llima, Ants Remm, Petr Zapletal, Nathan A. McMahon, Colin Scarato, Francois Swiadek, Christian Kraglund Andersen, Christoph Hellings, Sebastian Krinner, Nathan Lacroix, Stefania Lazar, Michael Kerschbaum, Dante Colao Zanuz, Graham J. Norris, Michael J. Hartmann, Andreas Wallraff, Christopher Eichler
TL;DR
Direct measurement of order parameters can require increased sampling near quantum phase boundaries, and state-preparation errors can reduce the measured string order. This work implements a QCNN on a seven-qubit superconducting processor to recognize SPT phases, and reports higher fidelity than direct string-order measurement for the prepared states.
Problem
Direct order-parameter measurements require increased sampling near phase boundaries, and hardware-prepared state errors can reduce the measured string order.
Source: p. 2Method
A QCNN designed to recognize string order processes approximate cluster-Ising ground states prepared and measured on a seven-qubit superconducting processor.
Source: pp. 2–3Results
The QCNN outperforms direct string-order measurement in correctly identifying the topological phase and tolerates X- and Z-type Pauli errors in weakly perturbed input states.
Source: pp. 2, 5Takeaways & Limitations
The QCNN output is equivalent to a weighted sum of string order parameters, allowing simultaneous measurement of terms whose number scales exponentially with system size.
Source: p. 5The demonstration is on a seven-qubit device; larger qubit number and circuit depth are described as future advances.
Source: pp. 3, 5
Abstract
from arXiv · showhide
Quantum computing crucially relies on the ability to efficiently characterize the quantum states output by quantum hardware. Conventional methods which probe these states through direct measurements and classically computed correlations become computationally expensive when increasing the system size. Quantum neural networks tailored to recognize specific features of quantum states by combining unitary operations, measurements and feedforward promise to require fewer measurements and to tolerate errors. Here, we realize a quantum convolutional neural network (QCNN) on a 7-qubit superconducting quantum processor to identify symmetry-protected topological (SPT) phases of a spin model characterized by a non-zero string order parameter. We benchmark the performance of the QCNN based on approximate ground states of a family of cluster-Ising Hamiltonians which we prepare using a hardware-efficient, low-depth state preparation circuit. We find that, despite being composed of finite-fidelity gates itself, the QCNN recognizes the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.
I. EXPERIMENTAL SETUP
The seven-qubit processor is fabricated on silicon and connected to cryogenic control and readout electronics. Flux lines tune qubit frequencies, while microwave and multiplexed readout hardware provide control and measurement.
Device fabrication
- The processor uses a niobium thin film on high-resistivity silicon, with airbridges for ground-plane connections and signal-line crossings.Aluminum Josephson junctions are fabricated by shadow evaporation.Source: §I · p. 6
Device parameters
- Qubit and readout-circuit parameters are extracted using spectroscopy and time-domain measurements.Measurement efficiencies are extracted using previously described methods.Source: §I · p. 6
Wiring and instrumentation
- At 13 mK, flux-control lines set qubit frequencies and nanosecond pulses activate two-qubit gates.A voltage-source offset sets idle frequencies; fast pulses from an arbitrary waveform generator activate gates.Source: §I · p. 6
- Two 8-channel AWGs drive the seven qubits, while FPGA-based systems perform frequency-multiplexed readout.Readout signals pass through amplification chains before digitization and integration.Source: §I · p. 6
II. CONTROL AND READOUT
The device uses microwave rotations and flux-pulsed CZ gates, with multiplexed single-shot readout of all seven qubits. Gate timing and readout procedures are designed to control leakage, population loss, and assignment errors.
Single-qubit gates
- Single-qubit Ry rotations use DRAG-shaped microwave pulses, with amplitudes scaled to set the target angle.The pulse width is 10 ns and its truncated length is 50 ns.Source: §II · p. 6
Two-qubit gates
- To activate a two-qubit controlled-Z (CZ) gate, both participating qubits are tuned to an intermediate interaction frequency where |11⟩ and non-computational |20⟩ become resonant.During idle times, the approximately 1.7 GHz detuning between two main frequency bands keeps residual ZZ coupling below 15 kHz for all qubit pairs.Source: §II · p. 7
- Net-zero transition-controlled flux pulses activate CZ gates; Gaussian filtering and 20 ns buffers help suppress SWAP errors and pulse overlap.The transition duration is fixed at 2.5 ns, and the average flux-pulse length is 71 ns.Source: §II · pp. 7–8, Figure 7 · p. 8
Multiplexed single-shot readout
- Simultaneous single-shot readout uses frequency-multiplexed pulses, integration weights, and thresholding to assign each qubit state.Assignment error increases with excitation number; prepared states in Fig. 8 are grouped by excitation count and ordered within each group.Source: Figure 8 · p. 8, §II · p. 9
III. GATE CHARACTERIZATION
Single-qubit randomized benchmarking and two-qubit process tomography characterize gate performance, while population-loss measurements help account for CZ gate infidelity. The measured single-qubit fidelity is high, and the most affected CZ gates are consistent with the population-loss estimates.
Single-qubit gate performance
- 99.82% average fidelity per Clifford was measured for single-qubit gates, with residual ZZ coupling below 15 kHz.Single-qubit fidelity was characterized by randomized benchmarking; residual coupling was measured using Ramsey experiments.Source: §III · p. 9, Figure 9 · p. 9
Two-qubit gate performance
- Quantum process tomography reconstructs each CZ process matrix from 16 input preparations and uses it to calculate gate infidelity against the ideal CZ process.Readout errors are corrected using the inverse assignment probability matrix before density-matrix reconstruction.Source: §III · p. 9
- CZ12 and CZ67 are most strongly affected by population loss during gate operation, consistent with their measured infidelities.Table II compares estimated bounds from measured population loss with gate infidelities measured by quantum process tomography.Source: Table II · p. 9, §III · p. 10
IV. METHODS
The methods combine classical variational optimization of state-preparation circuits with hardware execution and noise-aware simulation. Equivalent circuit parameters reduce excited-state occupation, while optimization near phase boundaries remains more challenging.
Variational circuit optimization
- Variational optimization achieved average energy accuracy ⟨H⟩−E0 ∼0.4 and average state infidelity 1−F ∼3.5%.Optimization typically converged within 1,000 iterations; Fig. 10 tracks energy and infidelity across runs.Source: §IV · p. 10
- Optimization failures above the 10% infidelity threshold occur more frequently near phase boundaries, where state infidelity and string-order deviations are also larger.Increasing circuit depth to m = 2 reduces infidelity, but the authors expect noise and gate errors to diminish that advantage and use m = 1.Source: §IV · p. 10
Exploiting ansatz symmetries
- Equivalent rotation angles can improve prepared-state fidelity by up to 10% compared with the original optimized angles.The equivalent parameters avoid first-layer rotations above π/2, reducing excited-state probability and thus amplitude damping.Source: §IV · p. 10
Ground state energy
- Measured ground-state energies reproduce the qualitative features of the exact energy diagram and agree with simulations that include decoherence and gate errors.The measurements cover 10×10 combinations of {h1, h2}.Source: §IV · pp. 10–11, Figure 12 · p. 11
Kraus operator simulations
- Kraus-operator simulations model state preparation and QCNN circuits as sequences of single-qubit, idle-qubit, and two-qubit gate processes.CZ processes are taken from experimental process tomography, with decoherence and residual ZZ coupling included for idle qubits.Source: §IV · p. 11
- The simulation models single-qubit Ry gates as concatenated rotation, residual ZZ-coupling, and relaxation/dephasing processes.The relaxation and dephasing parameters are taken from experimental characterization.Source: §IV · p. 11
V. PROPERTIES OF THE QCNN CIRCUIT
The QCNN uses cluster-state-inspired convolution and pooling to recognize the SPT phase, with a modified fully connected layer that addresses propagating Z errors. Its multi-scale string-order output can be sampled efficiently on the QCNN, whereas direct sampling of the modified observable becomes infeasible as depth grows.
QCNN circuit design
- The circuit is inspired by the MERA representation of the cluster state, which is constructed as a stable fixed point of the QCNN.Convolution and pooling map the seven-qubit cluster state to a reduced three-qubit cluster state.Source: §V · p. 11
- The modified fully connected layer addresses Z errors that propagate through convolution and pooling, while its measurement deterministically returns 1 for the ideal reduced cluster state and single Z errors on qubits 1, 4, or 7.Single X errors are absorbed by the convolutional and pooling layers.Source: §V · p. 12
Equivalent QCNN circuit
- An equivalent implementation reduces the full QCNN to two layers of CZ gates, projective measurement, and classical post-processing.The transformation moves conditional operations into post-processing and replaces a final controlled gate with measurements and a conditioned Z gate.Source: §V · pp. 12–13
Multi-scale string order parameter
- The QCNN’s multi-scale string order parameter contains O(256·3^(d−1)) products of O(32^d) different string order parameters, with maximum string length scaling as O(3^d).For the implemented depth-one, seven-qubit circuit, the finite system size limits the measured terms to 10.Source: §V · p. 13
Direct measurement of QCNN output
- The proposed QCNN’s first string-order component can be directly sampled with two measurement settings, but the modified circuit’s second component lacks that simple basis pattern.For depth d > 1, at least 16·3^(d−1) products in the second component require distinct measurement bases.Source: §V · p. 14
- For depth d > 1, at least 16·3^(d−1) products in the modified QCNN’s second string-order component require separate measurements, making direct sampling grow double-exponentially with depth.The QCNN instead samples the multi-scale string order parameter from one measurement of its output qubit.Source: §V · p. 14