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arXiv:2505.00683 · Physical Review A

Quantum circuit overhead

Oskar Słowik, Piotr Dulian, Adam Sawicki

Published May 18, 2026• 0 citations
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Abstract

We introduce a measure for evaluating the efficiency of finite universal quantum gate sets $\mathcal{S}$, called the Quantum Circuit Overhead (QCO), and the related notion of $T$-Quantum Circuit Overhead ($T$-QCO). QCO compares the circuit length required by $\mathcal S$ with the best possible length among gate sets of the same size. The $T$-QCO adapts this idea to cost models in which only selected costly gates are counted, while cheap operations are absorbed into an effective gate set. We demonstrate the usefulness of the ($T$-)QCO by extensive numerical calculations of its upper bounds, providing insight into the efficiency of various choices of single-qubit $\mathcal{S}$, including Haar-random gate sets and the gate sets derived from finite subgroups, such as Clifford and Hurwitz groups. In particular, our results suggest that, in terms of the upper bounds on the $T$-QCO, the famous T gate is a highly non-optimal choice for the completion of the Clifford gate set, even among the gates of order 8. We identify the optimal choices of such completions for both finite subgroups.

DOI
10.1103/vxx6-xtkz

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We introduce a measure for evaluating the efficiency of finite universal quantum gate sets S\mathcal{S}S, called the Quantum Circuit Overhead (QCO), and the related notion of TTT-Quantum Circuit Overhead (TTT-QCO). QCO compares the circuit length required by S\mathcal SS with the best possible length among gate sets of the same size. The TTT-QCO adapts this idea to cost models in which only selected costly gates are counted, while cheap operations are absorbed into an effective gate set. We demonstrate the usefulness of the (TTT-)QCO by extensive numerical calculations of its upper bounds, providing insight into the efficiency of various choices of single-qubit S\mathcal{S}S, including Haar-random gate sets and the gate sets derived from finite subgroups, such as Clifford and Hurwitz groups. In particular, our results suggest that, in terms of the upper bounds on the TTT-QCO, the famous T gate is a highly non-optimal choice for the completion of the Clifford gate set, even among the gates of order 8. We identify the optimal choices of such completions for both finite subgroups.
Among the cost metrics characterizing a quantum circuit, the T -count stands out as one of the most crucial as its minimization is particularly important in various areas of quantum computation such as fault-tolerant quantum computing and quantum circuit simulation. In this work, we contribute to the T -count reduction problem by proposing efficient T -count optimizers with low execution times. In particular, we greatly improve the complexity of TODD, an algorithm currently providing the best T -count reduction on various quantum circuits. We also propose some modifications to the algorithm which are leading to a significantly lower number of T gates. In addition, we propose another algorithm which has an even lower complexity and that achieves a better or equal T -count than the state of the art on most quantum circuits evaluated. We also prove that the number of T gates in the circuit obtained after executing our algorithms on a Hadamard-free circuit composed of n qubits is upper bounded by n ( n + 1 ) / 2 + 1 , which improves on the worst-case T -count of existing optimization algorithms. From this we derive an upper bound of ( n + 1 ) ( n + 2 h ) / 2 + 1 for the number of T gates in a Clifford + T circuit where h is the number of internal Hadamard gates in the circuit, i.e. the number of Hadamard gates lying between the first and the last T gate of the circuit.
We consider quantum circuits composed of Clifford and T gates. In this context the T gate has a special status since it confers universal computation when added to the (classically simulable) Clifford gates. However it can be very expensive to implement fault-tolerantly. We therefore view this gate as a resource which should be used only when necessary. Given an n-qubit unitary U we are interested in computing a circuit that implements it using the minimum possible number of T gates (called the T-count of U). A related task is to decide if the T-count of U is less than or equal to m; we consider this problem as a function of N=2^n and m. We provide a classical algorithm which solves it using time and space both upper bounded as O(N^m poly(m,N)). We implemented our algorithm and used it to show that any Clifford+T circuit for the Toffoli or the Fredkin gate requires at least 7 T gates. This implies that the known 7 T gate circuits for these gates are T-optimal. We also provide a simple expression for the T-count of single-qubit unitaries.
Quantum computers offer the potential to extend our abilities to tackle computational problems in fields such as number theory, encryption, search and scientific computation. Up to a superpolynomial speedup has been reported for quantum algorithms in these areas. Motivated by the promise of faster computations, the development of quantum machines has caught the attention of both academics and industry researchers. Quantum machines are now at sizes where implementations of quantum algorithms or their components are now becoming possible. In order to implement quantum algorithms on quantum machines, resource efficient circuits and functional blocks must be designed. In this work, we propose quantum circuits for Galois and integer arithmetic. These quantum circuits are necessary building blocks to realize quantum algorithms. The design of resource efficient quantum circuits requires the designer takes into account the gate cost, quantum bit (qubit) cost, depth and garbage outputs of a quantum circuit. Existing quantum machines do not have many qubits meaning that circuits with high qubit cost cannot be implemented. In addition, quantum circuits are more prone to errors and garbage output removal adds to overall cost. As more gates are used, a quantum circuit sees an increased rate of failure. Failures and error rates can be countered by using quantum error correcting codes and fault tolerant implementations of universal gate sets (such as Clifford+T gates). However, Clifford+T gates are costly to implement with the T gate being significantly more costly than the Clifford gates. As a result, designers working with Clifford+T gates seek to minimize the number of T gates (T-count) and the depth of T gates (T-depth). In this work, we propose quantum circuits for Galois and integer arithmetic with lower T-count, T-depth and qubit cost than existing work. This work presents novel quantum circuits for squaring and exponentiation over binary extension fields (Galois fields of form GF(2 m )). The proposed circuits are shown to have lower depth, qubit and gate cost to existing work. We also present quantum circuits for the core operations of multiplication and division which enjoy lower T-count, T-depth and qubit costs compared to existing work. This work also illustrates the design of a T-count and qubit cost efficient design for the square root. This work concludes with an illustration of how the arithmetic circuits can be combined into a functional block to implement quantum image processing algorithms.
We consider quantum circuit models where the gates are drawn from arbitrary gate ensembles given by probabilistic distributions over certain gate sets and circuit architectures, which we call stochastic quantum circuits. Of main interest in this work is the speed of convergence of stochastic circuits with different gate ensembles and circuit architectures to unitary t-designs. A key motivation for this theory is the varying preference for different gates and circuit architectures in different practical scenarios. In particular, it provides a versatile framework for devising efficient circuits for implementing ttt-designs and relevant applications including random circuit and scrambling experiments, as well as benchmarking the performance of gates and circuit architectures. We examine various important settings in depth. A key aspect of our study is an "ironed gadget" model, which allows us to systematically evaluate and compare the convergence efficiency of entangling gates and circuit architectures. Particularly notable results include i) gadgets of two-qubit gates with KAK coefficients (π4−18arccos⁡(15),π8,18arccos⁡(15))\left(\fracπ{4}-\frac{1}{8}\arccos(\frac{1}{5}),\fracπ{8},\frac{1}{8}\arccos(\frac{1}{5})\right)(4π​−81​arccos(51​),8π​,81​arccos(51​)) (which we call χχχ gates) directly form exact 2- and 3-designs; ii) the iSWAP gate family achieves the best efficiency for convergence to 2-designs under mild conjectures with numerical evidence, even outperforming the Haar-random gate, for generic many-body circuits; iii) iSWAP + complete graph achieve the best efficiency for convergence to 2-designs among all graph circuits. A variety of numerical results are provided to complement our analysis. We also derive robustness guarantees for our analysis against gate perturbations. Additionally, we provide cursory analysis on gates with higher locality and found that the Margolus gate outperforms various other well-known gates.
We consider quantum circuits composed of Clifford and TTT gates. In this context the TTT gate has a special status since it confers universal computation when added to the (classically simulable) Clifford gates. However it can be very expensive to implement fault-tolerantly. We therefore view this gate as a resource which should be used only when necessary. Given an nnn-qubit unitary UUU we are interested in computing a circuit that implements it using the minimum possible number of TTT gates (called the TTT-count of UUU). A related task is to decide if the TTT-count of UUU is less than or equal to mmm; we consider this problem as a function of N=2nN=2^nN=2n and mmm. We provide a classical algorithm which solves it using time and space both upper bounded as O(Nmpoly(m,N))\mathcal{O}(N^m \text{poly}(m,N))O(Nmpoly(m,N)). We implemented our algorithm and used it to show that any Clifford+T circuit for the Toffoli or the Fredkin gate requires at least 7 TTT gates. This implies that the known 7 TTT gate circuits for these gates are TTT-optimal. We also provide a simple expression for the TTT-count of single-qubit unitaries.
Abstract In quantum computing there are quite a few universal gate sets, each having their own characteristics. In this paper we study the Clifford+CS universal fault-tolerant gate set. The CS gate is used is many applications and this gate set is an important alternative to Clifford+T. We introduce a generating set in order to represent any unitary implementable by this gate set and with this we derive a bound on the CS-count of arbitrary multi-qubit unitaries. Analysing the channel representation of the generating set elements, we infer JnCS⊂JnT{\mathcal {J}}_n^{CS}\subset {\mathcal {J}}_n^TJnCS​⊂JnT​ J n CS ⊂ J n T , where JnCS{\mathcal {J}}_n^{CS}JnCS​ J n CS and JnT{\mathcal {J}}_n^TJnT​ J n T are the set of unitaries exactly implementable by the Clifford+CS and Clifford+T gate sets, respectively. We develop CS-count optimal synthesis algorithms for both approximately and exactly implementable multi-qubit unitaries. With the help of these we derive a CS-count-optimal circuit for Toffoli, implying JnTof=JnCS{\mathcal {J}}_n^{Tof}={\mathcal {J}}_n^{CS}JnTof​=JnCS​ J n Tof = J n CS , where JnTof{\mathcal {J}}_n^{Tof}JnTof​ J n Tof is the set of unitaries exactly implementable by the Clifford+Toffoli gate set. Such conclusions can have an important impact on resource estimates of quantum algorithms.