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arXiv:2403.14746

Bootstrapping shallow circuits

Bao, Ning, Gün Süer

Published Mar 21, 2024• 0 citations
arXivHTMLPDF

Abstract

Recently, a polynomial time classical algorithm has been found for learning the shallow representation of a unitary $U$ acting on $n$-qubits, by learning local inversions and then sewing them back together with ancilla qubits and SWAP gates. In this work, we bootstrap local inversion learning (LIL) to optimize quantum circuit depth by learning shallow representations for its sub-unitaries. We recursively cut circuits and apply the LIL algorithm to replace sub-circuits with their shallow representations, if it can be found by the algorithm. If not, we keep cutting until the optimization terminates, either by finding shallow representations or by reaching constant-depth sub-circuits. By replacing sub-circuits with their shallow representations, we hope to obtain some compression of the quantum circuit. Due to the binary search structure, the optimization algorithm has time complexity logarithmic in the depth of the original given circuit.

DOI
10.48550/arxiv.2403.14746

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Recently, a polynomial time classical algorithm has been found for learning the shallow representation of a unitary UUU acting on nnn-qubits, by learning local inversions and then sewing them back together with ancilla qubits and SWAP gates. In this work, we bootstrap local inversion learning (LIL) to optimize quantum circuit depth by learning shallow representations for its sub-unitaries. We recursively cut circuits and apply the LIL algorithm to replace sub-circuits with their shallow representations, if it can be found by the algorithm. If not, we keep cutting until the optimization terminates, either by finding shallow representations or by reaching constant-depth sub-circuits. By replacing sub-circuits with their shallow representations, we hope to obtain some compression of the quantum circuit. Due to the binary search structure, the optimization algorithm has time complexity logarithmic in the depth of the original given circuit.
Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown n-qubit shallow quantum circuit U (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of U. We also provide a polynomial-time classical algorithm for learning the description of any unknown n-qubit state | ψ ⟩ = U | 0n ⟩ prepared by a shallow quantum circuit U (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of | ψ ⟩. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.
This work considers a variation on the problem of learning shallow quantum circuits. Given query access to the circuit, as well as knowledge of its gate layout, we consider the task of learning the specific gates used in the circuit, producing an operationally-equivalent circuit matching this structure. Building on recent work for learning Haar random brickwork circuits, we identify a meta-algorithmic framework for learning broad classes of circuits based on iterative local gate inversions at the front and back of the circuit. We apply these techniques to study random, all-to-all, two-local circuits, and provide analytical and numerical evidence that this ensemble undergoes a sharp learnability transition at depth d∗∼log⁡2n+log⁡2log⁡2nd^* \sim \log_2 n + \log_2\log_2 nd∗∼log2​n+log2​log2​n in the large size limit, based on an analysis of lightcone growth. These results have implications for recently proposed quantum cryptographic schemes based on the difficulty of circuit learning, though there are important distinctions with respect to our setting that suggest avenues for future study.
We present the first computationally-efficient algorithm for average-case learning of shallow quantum circuits with many-qubit gates. Specifically, we provide a quasi-polynomial time and sample complexity algorithm for learning unknown QAC0^00 circuits -- constant-depth circuits with arbitrary single-qubit gates and polynomially many CZCZCZ gates of unbounded width -- with at most logarithmic ancilla, up to inverse-polynomially small error. Furthermore, we show that the learned unitary can be efficiently synthesized in poly-logarithmic depth. This work expands the family of efficiently learnable quantum circuits, notably since in finite-dimensional circuit geometries, QAC0^00 circuits require polynomial depth to implement.
Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation.
Abstract. Despite fundamental interests in learning quantum circuits, the existence of a computationally efficient algorithm for learning shallow quantum circuits remains an open question. Because shallow quantum circuits can generate distributions that are classically hard to sample from, existing learning algorithms do not apply. In this work, we present a polynomial-time classical algorithm for learning the description of any unknown [Formula: see text]-qubit shallow quantum circuit [Formula: see text] (with arbitrary unknown architecture) within a small diamond distance using single-qubit measurement data on the output states of [Formula: see text]. We also provide a polynomial-time classical algorithm for learning the description of any unknown [Formula: see text]-qubit state [Formula: see text] prepared by a shallow quantum circuit [Formula: see text] (on a 2D lattice) within a small trace distance using single-qubit measurements on copies of [Formula: see text]. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions. This circuit representation yields an optimization landscape that can be efficiently navigated and enables efficient learning of quantum circuits that are classically hard to simulate.
Unitary synthesis underlies universal quantum computation by translating unitary transformations into circuits of elementary gates. We show that for any budget m≥0m\geq 0m≥0 of ancilla, every nnn-qubit unitary can be exactly implemented using a quantum circuit consisting of O(4n)O(4^n)O(4n) CNOT and single-qubit gates in O(n+4n/(n+m))O(n+4^n/(n+m))O(n+4n/(n+m)) layers attaining the asymptotically optimal depth--ancilla tradeoff while simultaneously achieving optimal circuit size. The key idea is to encode computational basis states as wavefunctions sharing a Gaussian envelope, converting unitary action on amplitudes into coordinate transformations that admit shallow circuit implementations. The continuous encoding is then discretized, with approximation errors removed by exact residual correction. The depth-ancilla tradeoff follows by recursively applying the quantum Shannon decomposition, synthesizing its smaller unitary factors, and reusing ancillary qubits.