Controlled measurement, Hermitian conjugation and normalization in matrix-manipulation algorithms

TL;DR

This work tackles the efficiency bottleneck in measurement-based quantum matrix-manipulation algorithms caused by vanishing ancilla-success probabilities. It introduces controlled measurement, where a measurement on one ancilla is conditioned on the state of another, enabling single-run success and removing garbage without post-selection. In addition, it develops two encoding extensions: separating real and imaginary parts to handle Hermitian conjugation, and relaxing the global normalization constraint to broaden feasible encodings. These innovations are integrated into a matrix-multiplication framework, preserving linear-depth characteristics while expanding operability to complex matrices and conjugation tasks, with practical pathways to apply them to related quantum linear algebra algorithms.

Abstract

In this paper, we solve three important problems that are revealed, in particular, in matrix-manipulation algorithms. The principal novelty is introducing the concept of controlled measurement that solves the problem of small access probability to the desired state of ancilla and possesses several remarkable properties. We also introduce separate encoding of the real and imaginary parts of a complex matrix that allows to include the Hermitian conjugation into the list of matrix manipulations. Finally, we weaken the constraints on the modulus of matrix elements unavoidably imposed by the normalization condition for a pure quantum state. The controlled measurement together with both other extensions are implemented into the matrix multiplication algorithm. The appropriate circuits are presented.

Figures (6)

• Figure 1: Application of controlled measurement in the algorithm present in Sec.\ref{['Section:M']}.
• Figure 2: Replacement of ordinary measurement of ancilla state (left circuit) with the subroutine of controlled measurement (right circuit).
• Figure 3: The circuit for the Hermitian conjugation, $Z\equiv \sigma^{(z)}$.
• Figure 4: Replacement of matrix encoding via Eq.(\ref{['inst']}) with encoding via Eq.(\ref{['inst2']}) for organizing effective manipulations with complex matrices and weakening the constraint on the matrix elements.
• Figure 5: (a) The circuit for the matrix multiplication algorithm; the state $|\Psi_{out}\rangle$ is formed by the subsystems $R_1$, $C_2$, $K_1$ and $M_1$; $X\equiv \sigma^{(x)}$, $Z\equiv \sigma^{(z)}$; we omit subscripts in notations $W^{(j)}$ for the brevity; operator $W^{(2)}$ is a novelty in comparison with earlier algorithm in ZBQKW_arXive2024; (b) Notation for the multi-qubit CNOT.