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Quantum State Tomography with Conditional Generative Adversarial Networks

Shahnawaz Ahmed, Carlos Sánchez Muñoz, Franco Nori, Anton Frisk Kockum

arXiv:2008.03240v2quant-phcs.LG

TL;DR

Quantum-state characterization is challenging because linear-inversion reconstruction can fail with noisy measurements or an ill-conditioned sensing matrix. This paper introduces QST-CGAN, which outperforms iterative maximum-likelihood reconstruction on optical states in fidelity, convergence time, and measurement data, and reaches fidelity > 0.999 in about two orders of magnitude fewer iterations.

  • Problem

    Quantum-state characterization is challenging because linear-inversion reconstruction can fail with noisy measurements or a high-condition-number sensing matrix.

    Source: p. 2

  • Method

    QST-CGAN maps measurement statistics and operators to a density matrix using a conditional GAN with custom layers for representing quantum states.

    Source: pp. 1–2

  • Results

    On optical-state reconstruction, QST-CGAN outperforms iterative MLE in fidelity, convergence time, and measurement data; it reaches fidelity > 0.999 in about two orders of magnitude fewer iterations.

    Source: pp. 1–3

  • Takeaways & Limitations

    When pre-trained on simulated data for the expected states and noise, the generator can reconstruct a state in a single evaluation.

    Source: pp. 2–3

    Evaluation focuses on optical quantum states, although the authors state the approach can apply to any quantum system and observable.

    Source: p. 3

Abstract

from arXiv · show

Quantum state tomography (QST) is a challenging task in intermediate-scale quantum devices. Here, we apply conditional generative adversarial networks (CGANs) to QST. In the CGAN framework, two duelling neural networks, a generator and a discriminator, learn multi-modal models from data. We augment a CGAN with custom neural-network layers that enable conversion of output from any standard neural network into a physical density matrix. To reconstruct the density matrix, the generator and discriminator networks train each other on data using standard gradient-based methods. We demonstrate that our QST-CGAN reconstructs optical quantum states with high fidelity orders of magnitude faster, and from less data, than a standard maximum-likelihood method. We also show that the QST-CGAN can reconstruct a quantum state in a single evaluation of the generator network if it has been pre-trained on similar quantum states.

N = 32

The QST-CGAN reaches high reconstruction fidelity with far fewer data points than iMLE, beginning to outperform it around N = 32. It also reconstructs experimental noisy states from limited measurements and enables single-shot reconstruction after pre-training.

N = 32

  • Around N = 32 data points, QST-CGAN begins outperforming iMLE, reaches fidelities close to unity with fewer than 100 points, while iMLE requires about 1000 for good fidelity.The comparison uses a pure state (rank r = 1); Fig. 3(b) reports average fidelity after 1000 iterations across 10 randomly selected displacement sets.Source: Figure 3 · p. 4, N = 32 · p. 4
  • QST-CGAN’s convergence speed and per-iteration computational cost depend on the network architecture and training parameters.The iteration-based comparison averages results over 100 runs with randomly initialized QST-CGAN weights and iMLE starting density matrices.Source: N = 32 · p. 4

Experimental state reconstruction from parity measurements

  • Despite significant SPAM noise, QST-CGAN reconstructs an experimentally generated Wigner-negative resonator state well using only about 15% of the Wigner-function measurement data.The state was generated with a superconducting transmon qubit by applying a π SNAP operation to |0⟩ and |1⟩ of a coherent state |α = 1⟩.Source: N = 32 · p. 4

Single-shot reconstruction with pre-training

  • After pre-training on several thousand similar cat states, QST-CGAN can perform single-shot reconstructions.The pre-training set randomly varies |α| from 1 to 3 and includes superpositions of up to six coherent states.Source: N = 32 · p. 4

QST-CGAN

QST-CGAN uses custom neural-network layers to enforce density-matrix properties and outperform iMLE in the reported examples. Pre-training on classes of quantum states also enables high-fidelity, rapid reconstruction.

QST-CGAN

  • ∼0.9 fidelity with the full reconstruction was achieved using ∼600 Wigner-function data points for the Wigner-negative state.The full reconstruction used 4,281 points inside the dashed circle; points outside it were less reliable because of calibration problems at higher photon numbers.Source: Figure 4 · p. 4
  • ∼0.98 average fidelity across different cat states; pre-training enabled reconstruction four orders of magnitude faster than Fig. 3(a), with tens of further iterations improving fidelity.Figure 5 shows single-shot reconstructions of 200 cat states with |α| ∈ [1, 3] and up to six coherent states in superposition, using a pre-trained QST-CGAN.Source: QST-CGAN · p. 4, Figure 5 · p. 4
  • ∼100× fewer iterations and ∼10× fewer data points, while consistently achieving higher fidelity than iMLE in the reported examples.The overall comparison is reported in the conclusion.Source: QST-CGAN · pp. 4–5
  • Two custom layers enforce density-matrix properties and allow calculation of measurement expectation values.The conclusion identifies these layers as the basis of the QST-CGAN adaptation.Source: QST-CGAN · p. 4
  • The custom layers could extend to Transformers [72]; proposed directions include denoising via pre-training on simulated noisy data and adaptive tomography guided by discriminator mismatch [59] [67].These are prospective applications identified in the outlook.Source: QST-CGAN · p. 5

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