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arXiv:2510.20730 · EPJ Quantum Technology

Optimal constant-cost implementations of Clifford operations using global interactions

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

Published May 26, 2026• 0 citations
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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.

DOI
10.1140/epjqt/s40507-026-00520-z

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arXiv:2005.05581•Quantum•Feb 15, 2021•6 citations
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.
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 6 application of such Clifford entangling multiqubit gates to realize any sequence of Clifford operations of any length, without ancillae. Furthermore, we show that any sequence of CNOT gates of any length, can be replaced with 5 applications of such Clifford entangling multiqubit gates, without ancillae. We investigate the required qubit drive power that is associated with these implementations. Our work introduces a practical and computationally efficient algorithm to realize these compilations.
Abstract There are various gate sets that can be used to describe a quantum computation. A particularly popular gate set in the literature on quantum computing consists of arbitrary single-qubit gates and two-qubit CNOT gates. A CNOT gate is however not always the natural multi-qubit interaction that can be implemented on a given physical quantum computer, necessitating a compilation step that transforms these CNOT gates to the native gate set. An especially interesting case where compilation is necessary is for ion trap quantum computers, where the natural entangling operation can act on more than two qubits and can even act globally on all qubits at once. This calls for an entirely different approach to constructing efficient circuits. In this paper we study the problem of converting a given circuit that uses two-qubit gates to one that uses global gates. Our three main contributions are as follows. First, we find an efficient algorithm for transforming an arbitrary circuit consisting of Clifford gates and arbitrary phase gates into a circuit consisting of single-qubit gates and a number of global interactions proportional to the number of non-Clifford phases present in the original circuit. Second, we find a general strategy to transform a global gate that targets all qubits into one that targets only a subset of the qubits. This approach scales linearly with the number of qubits that are not targeted, in contrast to the exponential scaling reported in (Maslov and Nam 2018 New J. Phys. 20 033018). Third, we improve on the number of global gates required to synthesise an arbitrary n-qubit Clifford circuit from the 12n − 18 reported in (Maslov and Nam 2018 New J. Phys. 20 033018) to 6n − 8.
We consider quantum circuits composed of single-qubit operations and global entangling gates generated by Ising-type Hamiltonians. It is shown that such circuits can implement a large class of unitary operators commonly used in quantum algorithms at a very low cost-using a constant or effectively constant number of global entangling gates. Specifically, we report constant-cost implementations of Clifford operations with and without ancillae, constant-cost implementation of the multiply-controlled gates with linearly many ancillae, and an O(log^{*}(n)) cost implementation of the n-controlled single-target gates using logarithmically many ancillae. This shows a significant asymptotic advantage of circuits enabled by the global entangling gates.
Approximate synthesis of general single-qubit unitaries over the Clifford+T\sqrt{T}T​ gate set
For the standard Clifford+TTT gate set, deterministic, ancilla-free synthesis now attains the minimal TTT-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The T\sqrt{T}T​ gate rotates by half the angle of TTT, generating a finer lattice of implementable operations. Access to this magic state is expected to lower the cost of deterministic and ancilla-free synthesis of general single-qubit unitaries (Kliuchnikov et al., arXiv:2203.10064), but no direct Clifford+T\sqrt{T}T​ algorithm existed for this case. We provide one by extending the integer lattice-point enumeration method of Morisaki et al. We adopt a resource state cost model based on the magic-state catalysis approach of Gidney and Fowler (arXiv:1812.01238). On Haar-random targets synthesized to precisions ranging from ε=10−3\varepsilon=10^{-3}ε=10−3 to 10−810^{-8}10−8, the cost of Clifford+T\sqrt{T}T​ circuits scales as 2.4log⁡2(1/ε)2.4\log_2(1/\varepsilon)2.4log2​(1/ε) compared to 3.0log⁡2(1/ε)3.0\log_2(1/\varepsilon)3.0log2​(1/ε) for TTT-count-optimal Clifford+TTT circuits. Once a one-time catalyst state is amortized, the Clifford+T\sqrt{T}T​ circuits are never costlier than their Clifford+TTT counterparts.
Abstract Compiling quantum circuits to account for hardware restrictions is an essential part of the quantum computing stack. Circuit compilation allows us to adapt algorithm descriptions into a sequence of operations supported by real quantum hardware, and has the potential to significantly improve their performance when optimisation techniques are added to the process. One such optimisation technique is reducing the number of quantum gates that are needed to execute a circuit. For instance, methods for reducing the number of non-Clifford gates or CNOT gates from a circuit are an extensive research area that has gathered significant interest over the years. For certain hardware platforms such as trapped-ion quantum computers, we can leverage some of their special properties to further reduce the cost of executing a quantum circuit in them. In this work we use global interactions, such as the Global Mølmer–Sørensen (MS) gate present in trapped-ion hardware, to optimise and synthesise quantum circuits. We design and implement an algorithm that is able to compile an arbitrary quantum circuit into another circuit that uses global gates as the entangling operation, while optimising the number of global interactions needed. The algorithm is based on the ZX -calculus and uses a specialised circuit extraction routine that groups entangling gates into Global MS gates. We benchmark the algorithm in a variety of circuits, and show how it improves their performance under state-of-the-art hardware considerations in comparison to a naive algorithm and the Qiskit optimiser.
For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current procedures do not yet support individual assignment of base gate costs and many do not support extended sets of universal base gates. We analysed cost-optimal sequences using an exhaustive search based on Dijkstra’s pathfinding algorithm for the canonical Clifford+ T set of base gates and compared them to when additionally including Z -rotations from higher orders of the Clifford hierarchy. Two approaches of assigning base gate costs were used. First, costs were reduced to T -counts by recursively applying a Z -rotation catalyst circuit. Second, costs were assigned as the average numbers of raw (i.e. physical level) magic states required to directly distil and implement the gates fault-tolerantly. We found that the average sequence cost decreases by up to 54±3% when using the Z -rotation catalyst circuit approach and by up to 33±2% when using the magic state distillation approach. In addition, we investigated observed limitations of certain assignments of base gate costs by developing an analytic model to estimate the proportion of sets of Z -rotation gates from higher orders of the Clifford hierarchy that are found within sequences approximating random target gates.