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Alternating Layered Variational Quantum Circuits Can Be Classically Optimized Efficiently Using Classical Shadows

Afrad Basheer, Yuan Feng, Christopher Ferrie, Sanjiang Li

TL;DR

$Variational quantum algorithms (VQAs) on near-term devices face training and sample-cost challenges. The paper introduces ALSO, an alternating layered shadow optimization that uses classical shadows to evaluate and optimize VQAs on a classical computer, achieving exponential savings in input-state copies when the depth is $d=O(\log n)$. It demonstrates two practical applications—state preparation and quantum autoencoders—where ALSO matches or outperforms ideal infinite-copy training with far fewer copies, and shows substantial resource gains over standard VQA training in simulations. The approach enables efficient, reusable shadows and extends to broader trainable ansätze and calibration tasks, offering a practically impactful route to scalable quantum-classical hybrid learning.$

Abstract

Variational quantum algorithms (VQAs) are the quantum analog of classical neural networks (NNs). A VQA consists of a parameterized quantum circuit (PQC) which is composed of multiple layers of ansatzes (simpler PQCs, which are an analogy of NN layers) that differ only in selections of parameters. Previous work has identified the alternating layered ansatz as potentially a new standard ansatz in near-term quantum computing. Indeed, shallow alternating layered VQAs are easy to implement and have been shown to be both trainable and expressive. In this work, we introduce a training algorithm with an exponential reduction in training cost of such VQAs. Moreover, our algorithm uses classical shadows of quantum input data, and can hence be run on a classical computer with rigorous performance guarantees. We demonstrate 2--3 orders of magnitude improvement in the training cost using our algorithm for the example problems of finding state preparation circuits and the quantum autoencoder.

Alternating Layered Variational Quantum Circuits Can Be Classically Optimized Efficiently Using Classical Shadows

TL;DR

d=O(\log n)

Abstract

Variational quantum algorithms (VQAs) are the quantum analog of classical neural networks (NNs). A VQA consists of a parameterized quantum circuit (PQC) which is composed of multiple layers of ansatzes (simpler PQCs, which are an analogy of NN layers) that differ only in selections of parameters. Previous work has identified the alternating layered ansatz as potentially a new standard ansatz in near-term quantum computing. Indeed, shallow alternating layered VQAs are easy to implement and have been shown to be both trainable and expressive. In this work, we introduce a training algorithm with an exponential reduction in training cost of such VQAs. Moreover, our algorithm uses classical shadows of quantum input data, and can hence be run on a classical computer with rigorous performance guarantees. We demonstrate 2--3 orders of magnitude improvement in the training cost using our algorithm for the example problems of finding state preparation circuits and the quantum autoencoder.
Paper Structure (23 sections, 3 theorems, 12 equations, 6 figures, 1 table)

This paper contains 23 sections, 3 theorems, 12 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Background
  4. Preliminaries
  5. Quantum computing
  6. Classical shadows using Pauli basis measurements
  7. Variational quantum algorithms
  8. Alternating Layered Shadow Optimization
  9. Alternating Layered Ansatz
  10. Method
  11. Sample complexity
  12. Applications
  13. The state preparation problem
  14. Quantum autoencoder
  15. Simulation Results
  16. ...and 8 more sections

Key Result

Theorem 1

Sack22 Let $\rho \in \mathbb{C} ^ {2 ^ n}$ be a quantum state. Suppose $O_1, O_2, \dots, O_M \in \mathbb{C} ^ {2 ^ n}$ are $M$$k$-local observables. For any $\delta, \epsilon \in \left( 0, 1\right)$, let $T$ be any integer not smaller than $\frac{4 ^ {k + 1}}{\epsilon ^ 2} \cdot \log(\frac{2M}{\del

Figures (6)

  • Figure 1: An illustration of alternating layered ansatzes where the parameterized sub-circuit $S(\boldsymbol{\theta}_{32})$ is applied on the first and the last qubits. Here, $\boldsymbol{\theta}$ is an order $3$ tensor with each $\boldsymbol{\theta}_{ij}$ being vectors of real parameters.
  • Figure 2: The structure of $S(\boldsymbol{\gamma})$ used in the simulation. The two-qubit gate used here is the CNOT gate.
  • Figure 3: The structure of $W_O(\boldsymbol{\theta}) = U(\boldsymbol{\theta}) ^ {\dag} O U (\boldsymbol{\theta})$ where the blue box is a $1$-local observable, and all other boxes are $S$ sub-circuits. Except for the red ones, all other sub-circuits cancel each other out, resulting in $W_O(\boldsymbol{\theta})$ being $2d$-local.
  • Figure 4: Plot showing the time (in seconds) taken for a single function evaluation using ALSO. Here, $d = \lfloor \log n \rfloor$, $S$ is a $2$-qubit parameterized sub-circuit and $O$ is a $1$-qubit observable. Along with the execution times, we plot the function $(0.02n)^5$ to highlight the polynomial dependence of time on the number of qubits.
  • Figure 5: Simulation results for state preparation (a-c) and quantum autoencoder (d-f) using SPSA. Each graph corresponds to 5 instances of a problem. VQA-$K$ consumes $K$ copies (samples) per function evaluation while ALSO-$T$ consumes $T$ copies (samples) in total. In (a) and (c), we compare the performance of ALSO with standard VQA in the case of $8$-qubit problems. Here, VQA-$10$, VQA-$50$ and VQA-$100$ will consume $4.8 \times 10^5$ ($4.8 \times 10^5$), $2.4 \times 10^6$ ($2.4 \times 10^6$) and $4.8 \times 10^6$ ($4.8 \times 10^6$) copies (samples) respectively while ALSO-$10^5$ consumes only $10^5$ ($10^5$) copies (samples), and still outperforms VQA considerably. Continuing in the $8$-qubit scenario, In (b) and (d), we compare the performance of ALSO with the ideal VQA that consumes infinite copies, and we see that ALSO is able to almost match the results of this ideal VQA using a modest $5 \times 10^5$ ($5 \times 10^5$) copies (samples). In (c) and (f), we plot results of similar experiments carried out on $30$-qubit states. In this case, VQA consumes $5.4 \times 10^6$ ($1.8 \times 10^6$) and $5.4 \times 10^7$ ($1.8 \times 10^7$) copies (samples) respectively. Note that here, only iteration numbers that are multiples of $1000$ are plotted.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof