Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reducing C-NOT Counts for State Preparation and Block Encoding via Diagonal Matrix Migration

Zexian Li, Guofeng Zhang, Xiao-Ming Zhang

Abstract

Quantum state preparation and block encoding are versatile and practical input models for quantum algorithms in scientific computing. The circuit complexity of state preparation and block encoding frequently dominates the end-to-end gate complexity of quantum algorithms. We give algorithms with lower C-NOT counts for both the state preparation and block encoding. For a general $n$-qubit state, we improve the C-NOT count from Plesch-Brukner algorithm, proposed in 2011, from $(23/24)2^n$ to $(11/12)2^n$. For block encoding, our single-ancilla protocol for $2^{n-1}\times 2^{n-1}$ matrices uses the spectral norm as subnormalization and achieves a C-NOT count leading term $(11/48)4^n$. This result even exceeds the lower bound of $(1/4)4^n$ for $n$-qubit unitary synthesis. Further optimization is performed for low-rank matrices, which frequently arise in practical applications. Specifically, we achieve the C-NOT count leading term $(K+(11/12))2^n$ for a rank-$K$ matrix. Our approach builds upon the recursive block-ZXZ decomposition from Krol et al. and introduces a diagonal matrix migration technique based on the commutativity of the diagonal matrix and the uniformly controlled rotation about the $z$-axis to minimize the use of C-NOT gates.

Reducing C-NOT Counts for State Preparation and Block Encoding via Diagonal Matrix Migration

Abstract

Quantum state preparation and block encoding are versatile and practical input models for quantum algorithms in scientific computing. The circuit complexity of state preparation and block encoding frequently dominates the end-to-end gate complexity of quantum algorithms. We give algorithms with lower C-NOT counts for both the state preparation and block encoding. For a general -qubit state, we improve the C-NOT count from Plesch-Brukner algorithm, proposed in 2011, from to . For block encoding, our single-ancilla protocol for matrices uses the spectral norm as subnormalization and achieves a C-NOT count leading term . This result even exceeds the lower bound of for -qubit unitary synthesis. Further optimization is performed for low-rank matrices, which frequently arise in practical applications. Specifically, we achieve the C-NOT count leading term for a rank- matrix. Our approach builds upon the recursive block-ZXZ decomposition from Krol et al. and introduces a diagonal matrix migration technique based on the commutativity of the diagonal matrix and the uniformly controlled rotation about the -axis to minimize the use of C-NOT gates.
Paper Structure (23 sections, 6 theorems, 32 equations, 6 figures, 4 tables)

This paper contains 23 sections, 6 theorems, 32 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. State Preparation Optimization
  5. Single Ancilla Block Encoding Protocol with the Optimal Normalization Factor
  6. Single Ancilla Block Encoding Protocol for Low-rank Matrices
  7. Organization
  8. Preliminaries
  9. Definition
  10. Notation
  11. State Preparation via Diagonal Matrix Migration
  12. Unitary Synthesis
  13. Decoupling a $\mathrm{U}(2^{n})$ Operation into $n$-qubit Multiplexor Operations with $n\ge 3$
  14. Decoupling an $n$-qubit Multiplexor into $\mathrm{U}(2^{n-1})$ Gates and a Uniformly Controlled Rotation with $n\ge 2$
  15. Decoupling a $\mathrm{U}(4)$ Operation into Single-qubit Gates and C-NOT Gates Up to a Diagonal Matrix
  16. ...and 8 more sections

Key Result

Theorem 1

Given an $n$-qubit state, it can be prepared using at most $\frac{11}{12}2^n$ C-NOT gates. The precise C-NOT count given by $N_{{\rm state}}(n)$ (Eq. eq C-NOT state preparation) is less than $\frac{11}{12}2^n$.

Figures (6)

  • Figure 1: The process of synthesizing SPDMM. In the figure above, the green area is dedicated to processing the data, while the blue area is used for generating the quantum circuit. The algorithm breaks down the state preparation routine in each recursion into two subroutines: (1) Unitary/isometry synthesis subroutines generate $U$ and $\operatorname{conj}(V)$, up to diagonal matrices $\Delta_U$ and $\Delta_V$; (2) Another level recursion of state preparation subroutine prepares $\ket{\psi'} = \text{diag}(\Delta_U^\dagger \cdot \Sigma \cdot \operatorname{conj}(\Delta_V))$. The state preparation circuit can be recursively generated using the same process until $n=1$.
  • Figure 2: Quantum circuit of unitary and isometry synthesis by the block-ZXZ decomposition.
  • Figure 3: The circuit for preparing $2$-$5$ qubit states via SPDMM.
  • Figure 4: Quantum circuit of single ancilla block encoding for a matrix in $\mathbb{C}^{2^{n-1}\times 2^{n-1}}$ with spectral norm $\Vert A\Vert_2\leq 1$.
  • Figure 5: The process of synthesizing the SIABLE. The synthesis of $V_A^\dagger$ up to $\Delta_{V_A}$ will migrate into the synthesis of $W_A\cdot \Delta_{V_A}^\dagger$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Lemma 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5: C-NOT counts for encoding low-rank matrix
  • proof