Spacetime-Efficient Low-Depth Quantum State Preparation with Applications

Kaiwen Gui1,2,3, Alexander M. Dalzell4, Alessandro Achille5, Martin Suchara1, and Frederic T. Chong3

1Amazon Web Services, WA, USA
2Pritzker School of Molecular Engineering, University of Chicago, IL, USA
3Department of Computer Science, University of Chicago, IL, USA
4AWS Center for Quantum Computing, Pasadena, CA, USA
5AWS AI Labs, Pasadena, CA, USA

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Abstract

We propose a novel deterministic method for preparing arbitrary quantum states. When our protocol is compiled into CNOT and arbitrary single-qubit gates, it prepares an $N$-dimensional state in depth $O(\log(N))$ and $\textit{spacetime allocation}$ (a metric that accounts for the fact that oftentimes some ancilla qubits need not be active for the entire circuit) $O(N)$, which are both optimal. When compiled into the $\{\mathrm{H,S,T,CNOT}\}$ gate set, we show that it requires asymptotically fewer quantum resources than previous methods. Specifically, it prepares an arbitrary state up to error $\epsilon$ with optimal depth of $O(\log(N) + \log (1/\epsilon))$ and spacetime allocation $O(N\log(\log(N)/\epsilon))$, improving over $O(\log(N)\log(\log (N)/\epsilon))$ and $O(N\log(N/\epsilon))$, respectively. We illustrate how the reduced spacetime allocation of our protocol enables rapid preparation of many disjoint states with only constant-factor ancilla overhead – $O(N)$ ancilla qubits are reused efficiently to prepare a product state of $w$ $N$-dimensional states in depth $O(w + \log(N))$ rather than $O(w\log(N))$, achieving effectively constant depth per state. We highlight several applications where this ability would be useful, including quantum machine learning, Hamiltonian simulation, and solving linear systems of equations. We provide quantum circuit descriptions of our protocol, detailed pseudocode, and gate-level implementation examples using Braket.

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[1] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer, "Trading T gates for dirty qubits in state preparation and unitary synthesis", Quantum 8, 1375 (2024).

[2] Xiao-Ming Zhang and Xiao Yuan, "Circuit complexity of quantum access models for encoding classical data", npj Quantum Information 10 1, 42 (2024).

[3] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[4] Raghav Jumade and Nicolas PD Sawaya, "Data is often loadable in short depth: Quantum circuits from tensor networks for finance, images, fluids, and proteins", arXiv:2309.13108, (2023).

[5] Erik J. Gustafson, Yao Ji, Henry Lamm, Edison M. Murairi, and Shuchen Zhu, "Primitive Quantum Gates for an SU(3) Discrete Subgroup: $\Sigma(36\times3)$", arXiv:2405.05973, (2024).

[6] Gideon Lee, Connor T. Hann, Shruti Puri, S. M. Girvin, and Liang Jiang, "Error Suppression for Arbitrary-Size Black Box Quantum Operations", Physical Review Letters 131 19, 190601 (2023).

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[8] Vu Tuan Hai, Nguyen Tan Viet, and Le Bin Ho, "〈qo|op〉: A quantum object optimizer", SoftwareX 26, 101726 (2024).

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