Quantum Resources Required to Block-Encode a Matrix of Classical Data

Abstract

We provide a modular circuit-level implementation and resource estimates for several methods of block-encoding a dense N × N matrix of classical data to precision ; the minimal-depth method achieves a T -depth of O(log(N/ )), while the minimal-count method achieves a T -count of O(N log(1/ )). We examine resource tradeoffs between the different approaches, and we explore implementations of two separate models of quantum random access memory (QRAM). As part of this analysis, we provide a novel state preparation routine with T -depth O(log(N/ )), improving on previous constructions with scaling O(log 2 (N/ )). Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms. INDEX TERMSQuantum computing, quantum circuits, block encoding, quantum random access memory, quantum circuit synthesis, quantum resources I. INTRODUCTION A. MOTIVATION

Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra

“…This algorithm only needs to make a small number of calls to the block encoding if the matrix is well-conditioned, compared to the runtime of classical linear systems algorithms. However, for general matrices an implementation of the block encoding using QRAM 2 in minimum depth O(log(N )) requires O(N 2 ) qubits [12,15,16]. 3 Thus, for general matrices the exponential savings in space resources are nullified.…”

“…We assume a Frobenius norm block encoding, with explicit construction via a QRAM in minimal depth as detailed in Ref. [16]. We note there are also other approaches which are more space efficient in quantum resources, but more costly in quantum runtime.…”

“…Quantum Random Access Memory (QRAM): a widely-studied data access structure that allows for coherent access to classical data[12][13][14] 3 The minimum qubit count implementation in Ref [16]. requires O(N ) qubits and O(N ) depth per call to the block encoding.…”

Circuit complexity of quantum access models for encoding classical data

“…In particular, ref. 16 presents a nearly timeoptimal protocol for block-encoding of general dense matrices of 2 n × 2 n dimension. A circuit depth of Õ ðnÞ can be achieved at the expense of exponential ancillary qubits.…”

Robustness of Gauge Digitization to Quantum Noise

“…Different layouts and mixed architectures (combining qubits and qudits) are being investigated [20][21][22][23][24][25][26][27][28][29]. Further questions persist around the merits of quantum memory such as qRAM [30][31][32][33][34][35][36] and efficient quantum error correction (QEC) [37][38][39][40] where between O(10 1−5 ) physical qubits per logical qubit appear necessary for fault tolerance [41][42][43][44]. Meanwhile, quantum advantage for HEP problems may be possible by designing them to be robust to certain errors, using partial QEC [45][46][47][48][49][50][51], hardware-aware embeddings [52][53][54], or biased-noise qubits [55,56].…”