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Quantum process tomography with unsupervised learning and tensor networks

Giacomo Torlai, Christopher J. Wood, Atithi Acharya, Giuseppe Carleo, Juan Carrasquilla, Leandro Aolita

arXiv:2006.02424v1quant-ph

TL;DR

Quantum process tomography requires exponentially scaling data and classical post-processing, limiting the scale of full characterization. This paper combines a tensor-network representation of the channel with data-driven optimization inspired by unsupervised learning. In simulations, it reconstructs random circuits up to 10 qubits and noisy five-qubit circuits with process fidelities close to 0.99.

  • Problem

    Full quantum process tomography requires exponentially scaling data and classical post-processing, and scalable reconstruction of noisy quantum circuits remains an open question.

    Source: p. 1

  • Method

    The approach represents the Choi matrix as a locally purified tensor network and optimizes its parameters by gradient descent against measurement data.

    Source: pp. 2–3

  • Results

    Simulated reconstructions reach fidelity above 0.99 for 10-qubit depth-4 random circuits with fewer than 10^5 single-shot measurements, while noisy-channel reconstructions approach 0.99.

    Source: pp. 2, 5

  • Takeaways & Limitations

    The simulations demonstrate QPT on random circuits up to 10 qubits and a noisy five-qubit circuit, with resulting process fidelities close to 0.99.

    Source: p. 5

    The LPDO parameterization is not trace-preserving by construction; trace preservation is expected only if the model faithfully learns the channel and can be approximately encouraged with regularization.

    Source: p. 3

Abstract

from arXiv · show

The impressive pace of advance of quantum technology calls for robust and scalable techniques for the characterization and validation of quantum hardware. Quantum process tomography, the reconstruction of an unknown quantum channel from measurement data, remains the quintessential primitive to completely characterize quantum devices. However, due to the exponential scaling of the required data and classical post-processing, its range of applicability is typically restricted to one- and two-qubit gates. Here, we present a new technique for performing quantum process tomography that addresses these issues by combining a tensor network representation of the channel with a data-driven optimization inspired by unsupervised machine learning. We demonstrate our technique through synthetically generated data for ideal one- and two-dimensional random quantum circuits of up to 10 qubits, and a noisy 5-qubit circuit, reaching process fidelities above 0.99 using only a limited set of single-qubit measurement samples and input states. Our results go far beyond state-of-the-art, providing a practical and timely tool for benchmarking quantum circuits in current and near-term quantum computers.

I. QUANTUM CHANNELS

A quantum channel is a completely positive, trace-preserving map, expressible through Kraus operators or as a Choi matrix. The Choi representation encodes the channel’s input and output spaces and carries equivalent CP and TP constraints.

I. QUANTUM CHANNELS

  • A quantum channel maps a density operator to another through a completely positive, trace-preserving linear map.In the Kraus representation, the operators satisfy a completeness relation.Source: §I · pp. 8–9
  • The Choi matrix is obtained by applying the channel to one half of a tensor product of unnormalized Bell pairs.It is a positive-semidefinite operator on an extended space of input and output qubits, normalized to trace d^N.Source: §I · p. 9
  • The channel is trace-preserving exactly when tracing the Choi matrix over physical, or output, qubits gives the identity on ancillary, or input, qubits.The input and output spaces also provide the interpretation of the Choi matrix’s ancillary and physical degrees of freedom.Source: §I · p. 9

Tensor networks for Choi matrices

Tensor networks can represent a Choi matrix compactly when the channel has structure, such as a low-depth unitary circuit. For unitary circuits, rearranging the Bell-pair indices avoids the large bond dimension of the canonical ordering.

Tensor networks for Choi matrices

  • 38: The Choi matrix is a rank-4N tensor, and exact classical representations of quantum states and operators scale exponentially with system size.Its canonical form places input and output indices on opposite halves of the tensor.Source: Tensor networks for Choi matrices · p. 9
  • A unitary circuit can admit an efficient MPO representation depending on its gates, geometry, and depth.The MPO bond dimension χU measures the representation’s complexity.Source: Tensor networks for Choi matrices · p. 9
  • Canonical index swapping makes the Bell-pair MPS maximally entangled across 2N qubits, saturating its bond dimension at 2^N.This is an inefficient representation for constructing the Choi matrix.Source: Tensor networks for Choi matrices · p. 9
  • Contracting the unitary MPO with the Bell-pair physical indices and folding the ancillary indices back yields a rank-1 Choi matrix as an MPS.Its physical dimension is d^2 and its bond dimension is χU, so construction is efficient when χU is sufficiently low.Source: Tensor networks for Choi matrices · p. 10

II. QUANTUM PROCESS TOMOGRAPHY

Quantum process tomography reconstructs a channel by fitting its Choi matrix to input-state and output-measurement data. The standard full-matrix approach requires exponentially scaling resources, while informationally complete data can uniquely characterize the channel.

II. QUANTUM PROCESS TOMOGRAPHY

  • An informationally complete set of inputs and measurements spans the channel’s input and output spaces, and its outcome probabilities uniquely characterize the channel.Those probabilities can be used to reconstruct the unknown Choi matrix.Source: §II · p. 10
  • A training sample consists of an input choice α and a measurement outcome β drawn from the channel’s process probability distribution.The measurement outcome can be sampled one qubit at a time by sweeping across conditional probabilities.Source: Figure 7 · p. 11, §II · p. 11
  • The standard approach parametrizes the full Choi matrix and estimates its elements by maximum likelihood, subject to complete-positivity and trace-preserving constraints.The constraints are ΛE ≥ 0 and Trτ ΛE = 1σ.Source: §II · p. 11
  • Full QPT requires exponentially scaling matrix parameters and the full informationally complete set of inputs and measurements, restricting it to very small systems.Both the data requirements and classical post-processing scale exponentially with qubit count.Source: §II · p. 11

Data sets generation

The data-generation procedure selects product input states from a single-qubit POVM and samples output measurement outcomes from the resulting process probabilities. For unitary circuits it builds an MPO without discarding nonzero singular values; noisy five-qubit data are generated from a directly simulated Choi matrix.

Data sets generation

  • For the Pauli-projector POVM, six single-qubit outcomes define product input states, with equal probabilities assigned to the x, y, and z bases.The input labels form a string α specifying the product state.Source: Data sets generation · p. 11
  • The unitary circuit is compiled into an MPO by contracting gates and restoring the MPO form with SVD, discarding only zero singular values.Thus the MPO is not approximated in this step, while its bond dimension generally grows exponentially with circuit depth.Source: Data sets generation · pp. 11–12
  • Each output measurement outcome is sampled from the process probability by successively calculating conditional probabilities across qubits.The procedure repeats the contraction and sampling step until it produces a complete outcome β.Source: Data sets generation · p. 12
  • For noisy channels, training data are obtained from a directly simulated full Choi matrix for the five-qubit systems studied.The input states and measurement operators match those used for the unitary-circuit data.Source: Data sets generation · p. 12
  • An LPDO groups each site’s input and output indices into a local tensor representation of the Choi matrix.Setting the Kraus dimension to χν = 1 reduces it to a rank-1 Choi matrix represented by an MPS.Source: Figure 8 · p. 12

Locally-purified density operators

The channel’s Choi matrix is represented as a locally-purified density operator (LPDO) built from tensors with input, output, bond, and Kraus indices. Its construction ensures positivity and Hermiticity, while the Kraus dimension controls whether it is rank one.

Locally-purified density operators

Unsupervised learning

The LPDO parameters are optimized against process measurement data using a probability-based objective. Tensor-network contractions and stored environment tensors support the computation of normalization and data-dependent gradients.

Unsupervised learning

  • The LPDO parameters are variationally optimized by minimizing the Kullback-Leibler divergence, yielding a negative log-likelihood objective after averaging over the data set.Source: Unsupervised learning · pp. 12–13
  • Stored left- and right-environment tensors support sweeps that compute the normalization and its gradients across the LPDO.The environments are computed in opposite directions and reused to evaluate tensor-component gradients.Source: Figure 9 · p. 13, Unsupervised learning · pp. 13–14
  • The model probability Pϑ(β | α) is obtained by normalizing its unnormalized probability, linking the LPDO Choi matrix to process measurement outcomes.Source: Unsupervised learning · p. 13
  • For each data point, the LPDO is contracted with its input state and measurement operator to compute an unnormalized probability and its gradient; final gradients are averaged over samples.Source: Figure 9 · p. 13, Unsupervised learning · p. 14

Trace-preserving regularization

The cost function can penalize deviations from trace preservation alongside the negative log-likelihood. During training, the regularization measurement correlates with reconstruction infidelity.

Trace-preserving regularization

  • An added cost-function term biases optimization to minimize both the negative log-likelihood and the distance between TrτΛϑ and the identity on the input space.The distance uses the Frobenius norm of Δϑ = TrτΛϑ − 1σ.Source: Trace-preserving regularization · p. 14
  • The deviation tensor Δϑ is computed by subtracting the identity MPO, increasing the LPDO’s bond dimension by one.Source: Trace-preserving regularization · p. 14
  • The regularization Γϑ correlates with reconstruction infidelity during training for one-dimensional random circuits of different depths.Figure 10 shows the regularization and infidelity during training for a 10-qubit, depth-2 circuit at different data-set sizes.Source: Figure 10 · p. 14, Trace-preserving regularization · p. 14

Gradient updates

The LPDO parameters are updated by gradient descent, with minibatches used to reduce the slowness of full-data gradient updates. Adam adapts the learning rate using moving estimates of gradient moments.

Gradient updates

  • Full-data gradient updates can train very slowly when the number of samples M is large.Source: Gradient updates · p. 14
  • Minibatch gradients use MB ≪ M samples, enabling faster training; their fluctuations also help optimization escape local minima.An epoch sweeps the reshuffled full data set, updating parameters M/MB times.Source: Gradient updates · p. 15
  • Adam assigns each parameter an adaptive learning rate and estimates moving averages of first and second gradient moments, with ξ1 and ξ2 controlling their decay.Source: Gradient updates · p. 15

Overfitting and model selection

LPDO training must generalize beyond its training measurements; a held-out validation set helps identify overfitting and select model parameters without using information about the channel.

Overfitting and model selection

  • The data are split 80%/20% into training and validation sets, with gradients computed from training data and losses evaluated on both sets.The validation set is held out from the learning procedure.Source: Overfitting and model selection · p. 15
  • An increasing validation loss signals overfitting to spurious training features and indicates that more training data is needed.Early in training, validation and training losses tend to decrease together.Source: Overfitting and model selection · p. 15
  • The selected parameters are those from the epoch with the lowest validation loss, a procedure also usable in experiments.Source: Overfitting and model selection · p. 15

Fidelity estimation

The reconstruction is assessed using quantum process fidelity, defined through the state fidelity between Choi matrices. The metric is not scalable to large systems and requires a target Choi matrix that may not be known in practice.

Fidelity estimation

  • Quantum process fidelity is the quantum state fidelity between the learned and target Choi matrices.Source: Fidelity estimation · p. 15
  • The measurement is restricted to small system sizes and is not useful when the target Choi matrix is unknown.In practice, reconstruction fidelity is computed using the target Choi matrix for the circuit’s ideal unitary.Source: Fidelity estimation · p. 15
  • For a rank-1 LPDO Choi matrix, the fidelity expression reduces to a special case.Source: Fidelity estimation · p. 15

Specifics of the numerical experiments

The numerical experiments vary circuit and LPDO settings, use cross-validation for model selection, and examine sample complexity and reconstruction behavior. They include ideal circuit simulations and a decay channel applied at each gate.

Specifics of the numerical experiments

  • LPDO tensors are randomly initialized, and gradients are computed on batches of 800 samples before Adam updates.The stated learning rate is η = 0.005.Source: Specifics of the numerical experiments · p. 16
  • Sample complexity is measured by increasing the sample count until infidelity reaches ε = 0.025, then recording M* with an error bar set by the sample-size step.Source: Specifics of the numerical experiments · p. 16
  • The two-dimensional circuit MPO uses a snake-shaped representation, restored to local form by SVD while discarding only zero singular values, so the target representation remains exact.The circuit setup also specifies alternating odd- and even-index controls for CX layers and matches the LPDO bond dimension to the circuit MPO.Source: Specifics of the numerical experiments · p. 16
  • For the Fig. 3 study, the LPDO bond dimension matches the circuit MPO bond dimension and χν = 1; cross-validation selects models for each configuration and sample size.Simulations with larger LPDO bond dimensions gave comparable results.Source: Specifics of the numerical experiments · p. 16
  • The noisy circuit applies a decay-probability channel at each gate, using the tensor product of the single-qubit channel for two-qubit gates.Source: Specifics of the numerical experiments · p. 16
  • For a noiseless channel with γ = 0 and χν = 6, the reconstructed LPDO Choi-matrix purity reaches TrΛϑ ≈1 at M = 5 × 10^5 samples.Training metrics also converge for χν = 0 only in the noiseless setting.Source: Specifics of the numerical experiments · p. 16
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