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Optimal constant-cost implementations of Clifford operations using global interactions

Jonathan Nemirovsky, Lee Peleg, Amit Ben Kish, Yotam Shapira

TL;DR

The paper shows that arbitrary Clifford operations can be implemented with single-qubit gates and at most four multiqubit programmable all-to-all entangling gates, saturating the constant-cost lower bound. It provides an explicit decomposition that reduces the Clifford circuit to a sequence starting and ending with edge MQ gates, enabling merging with correlated rotation layers to achieve a four-gate realization. A symplectic-formalism framework underpins the construction, with a constructive CX decomposition into four MQ gates via symmetric matrices $E_1$, $E_2$, $F$, and $C=E_1^{-1}E_2$, plus a drive-power analysis showing comparable nuclear-norm requirements to standard methods and a favorable scaling. The results offer a practical, efficient compilation method for trapped-ion and related platforms leveraging all-to-all interactions, with a computable algorithm for real-time realizations.

Abstract

We investigate quantum circuits built from arbitrary single-qubit operations combined with programmable all-to-all multiqubit entangling gates that are native to, among other systems, trapped-ion quantum computing platforms. We report a constant-cost of no more than four applications of such Clifford entangling multiqubit gates to realize any sequence of Clifford operations of any length, without ancillae, which is the theoretically optimal gate count cost. We do this by implementing any sequence of CNOT gates of any length with four applications of such gates, without ancillae, and show that the extension to general Clifford operations incurs no additional cost. We investigate the required qubit drive power that is associated with our implementation and show that it is lower than that of a standard approach. Our work introduces a practical and computationally efficient algorithm to realize these compilations.

Optimal constant-cost implementations of Clifford operations using global interactions

TL;DR

The paper shows that arbitrary Clifford operations can be implemented with single-qubit gates and at most four multiqubit programmable all-to-all entangling gates, saturating the constant-cost lower bound. It provides an explicit decomposition that reduces the Clifford circuit to a sequence starting and ending with edge MQ gates, enabling merging with correlated rotation layers to achieve a four-gate realization. A symplectic-formalism framework underpins the construction, with a constructive CX decomposition into four MQ gates via symmetric matrices , , , and , plus a drive-power analysis showing comparable nuclear-norm requirements to standard methods and a favorable scaling. The results offer a practical, efficient compilation method for trapped-ion and related platforms leveraging all-to-all interactions, with a computable algorithm for real-time realizations.

Abstract

We investigate quantum circuits built from arbitrary single-qubit operations combined with programmable all-to-all multiqubit entangling gates that are native to, among other systems, trapped-ion quantum computing platforms. We report a constant-cost of no more than four applications of such Clifford entangling multiqubit gates to realize any sequence of Clifford operations of any length, without ancillae, which is the theoretically optimal gate count cost. We do this by implementing any sequence of CNOT gates of any length with four applications of such gates, without ancillae, and show that the extension to general Clifford operations incurs no additional cost. We investigate the required qubit drive power that is associated with our implementation and show that it is lower than that of a standard approach. Our work introduces a practical and computationally efficient algorithm to realize these compilations.
Paper Structure (1 section, 18 equations, 1 figure)

This paper contains 1 section, 18 equations, 1 figure.

Table of Contents

  1. Step-by-step derivation of Eq. (\ref{['eqMagic']})

Figures (1)

  • Figure 1: Total nuclear norm (vertical) of decompositions of random $-\text{CX}-$ layers over a varying number of qubits, $n$ (horizontal), with standard Gaussian elimination (orange) and with our constant-cost implementation (blue). Data (markers) are fitted to a power-law (solid) with $\Omega_\text{nuc}\propto n^\beta$, with the values of $\beta$ and confidence intervals shown in the legend.