Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis

Abstract

The Quantum State Preparation problem aims to prepare an n-qubit quantum state |ψ v = 2 n −1 k=0 v k |k from initial state |0 ⊗n , for a given unit vectorThe problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth complexity remains open in the general case with ancillary qubits. In this paper, we study efficient constructions of quantum circuits for preparing a quantum state: Given m = O(2 n /n 2 ) ancillary qubits, we construct a circuit to prepare |ψ v with depth Θ(2 n /(m + n)), which is optimal in this regime. In particular, when m = Θ(2 n /n 2 ), the circuit depth is Θ(n 2 ), which is an exponential improvement of the previous bound of O(2 n ). For m = ω(2 n /n 2 ), we prove a lower bound of Ω(n), an exponential improvement over the previous lower bound of Ω(log n), leaving only a polynomial gap between Ω(n) and O(n 2 ) for the depth complexity. These results also imply a tight bound of Θ(4 n /(m+n)) for depth of circuits implementing a general n-qubit unitary using m = O(2 n /n) ancillary qubits. This closes a gap for circuits without ancillary qubits; for circuits with sufficiently many ancillary qubits, this gives a quadratic saving from O(4 n ) to Θ(2 n ).Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide range of parameter regime. The results can be viewed as (optimal) time-space tradeoff bounds, which is not only theoretically interesting, but also practically relevant in the current trend that the number of qubits starts to take off, by showing a way to use a large number of qubits to compensate the short qubit lifetime.

Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications

“…The speed limit of quantum state preparation is a question with fundamental and practical interests, determining the efficiency of inputting classical data into a quantum computer, and playing as a critical subroutine for many quantum algorithms, such as in machine learning [1][2][3] and Hamiltonian simulations [4,5]. Without ancillary qubit, an exponential circuit depth is inevitable to prepare an arbitrary quantum state [6][7][8][9][10][11][12][13][14][15][16] and the optimal result Θ(2 𝑛 /𝑛) is recently obtained by Sun et al [17]. Leveraging ancillary qubits, the circuit depth could be reduced to be sub-exponential scaling [17][18][19][20], yet in the worse case with an exponential number of ancillas.…”

“…Without ancillary qubit, an exponential circuit depth is inevitable to prepare an arbitrary quantum state [6][7][8][9][10][11][12][13][14][15][16] and the optimal result Θ(2 𝑛 /𝑛) is recently obtained by Sun et al [17]. Leveraging ancillary qubits, the circuit depth could be reduced to be sub-exponential scaling [17][18][19][20], yet in the worse case with an exponential number of ancillas. Very recently, the optimal circuit depth Θ(𝑛) was achieved by [17,20] with 𝑂 (2 𝑛 ) [17] and Õ (2 𝑛 ) [20] ancillary qubits.…”

“…Leveraging ancillary qubits, the circuit depth could be reduced to be sub-exponential scaling [17][18][19][20], yet in the worse case with an exponential number of ancillas. Very recently, the optimal circuit depth Θ(𝑛) was achieved by [17,20] with 𝑂 (2 𝑛 ) [17] and Õ (2 𝑛 ) [20] ancillary qubits.…”

“…We first develop a deterministic algorithm (independent of Refs. [17,20]) for preparing an arbitrary quantum state with optimal circuit depth Θ(𝑛) and 𝑂 (2 𝑛 ) ancillary qubits. We next develop an algorithm for 𝑑-sparse quantum states (𝑑 2) that achieves the optimal circuit depth Θ(log(𝑛𝑑)), exponentially faster than the best-known results [25][26][27].…”

“…Our method saturates the circuit depth lower bound Ω(𝑛) [17,18]. Below we sketch our protocol with five stages and refer to [31] for the formal description.…”