Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits

Abstract

As a cornerstone for many quantum linear algebraic and quantum machine learning algorithms, QRAM aims to provide the transformation of |i |0 n → |i |ψ i for all i ∈ {0, 1} k for the given nqubit states |ψ i . In this paper, we construct a quantum circuit for implementing QRAM, with depthn+k+m and size O 2 n+k for any given number m of ancillary qubits. These bounds, which can also be viewed as time-space tradeoff for the transformation, are optimal for any integer parameters m, k ≥ 0 and n ≥ 1.Our construction induces a circuit for the standard quantum state preparation (QSP) problem of |0 n → |ψ for any |ψ , pinning down its depth complexity to Θ(n+2 n /(n+m)) and its size complexity to Θ(2 n ) for any m. This improves results in a recent line of investigations of QSP with ancillary qubits.Another fundamental problem, unitary synthesis, asks to implement a general n-qubit unitary by a quantum circuit. Previous work shows a lower bound of Ω(n + 4 n /(n + m)) and an upper bound of O(n2 n ) for m = Ω(2 n /n) ancillary qubits. In this paper, we quadratically shrink this gap by presenting a quantum circuit of depth of O n2 n/2

Quantum algorithm for finding minimum values in a Quantum Random Access Memory

“…Random Access Memory (RAM) is a versatile, short-term memory used in computing for storing and retrieving information via bits [1]. Similarly, the concept of Quantum RAM (QRAM) emerges with the same goal but employing qubits to apply a superposition of states to achieve more efficient results for computational applications, whether quantum or classical [2][3][4][5][6][7][8][9][10][11]. Several works discuss the potential of its applications to optimize the execution of quantum algorithms, including quantum searching on a classical database [10,[12][13][14][15][16], collision finding [12,[17][18][19], and algorithms for solving linear systems [20][21][22][23].…”

Quantum algorithm for finding minimum values in a Quantum Random Access Memory

“…Random Access Memory (RAM) is a versatile, short-term memory used in computing for storing and retrieving information via bits [1]. Similarly, the concept of Quantum RAM (QRAM) emerges with the same goal but employing qubits to apply a superposition of states to achieve faster results for computational applications, whether quantum or classical [2][3][4][5][6][7][8][9][10][11]. Several works discuss the potential of its applications to optimize the execution of quantum algorithms, including quantum searching on a classical database [10,[12][13][14][15], collision finding [12,[16][17][18], and algorithms for solving linear systems [19][20][21][22], for instance.…”