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.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.

arXiv:1308.4134Aug 19, 201332 citations

On the Computational Power of Quantum Circuits with Clifford and π/8 Gates

Ken Matsumoto, Kazuyuki Amano

IEICE Technical Report; IEICE Tech. Rep.Mar 3, 20080 citations

Fast Equivalence Checking of Quantum Circuits of Clifford Gates

Dimitrios Thanos, Tim J. Coopmans, Alfons Laarman

arXiv:2308.01206Lecture notes in computer scienceJan 1, 202311 citations

An algorithm for the T-countAn algorithm for the T-count

David Gosset, Vadym Kliuchnikov, Michele Mosca +1 more

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 $\mathcal{O}(N^m \text{poly}(m,N))$ . 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.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 $\mathcal{O}(N^m \text{poly}(m,N))$. 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 Information and ComputationNov 1, 201465 citations

Resource-Efficient Synthesis of Sparse Quantum States

Renaud Vilmart, Sunheang Ty, Chetra Mang

Preparing a quantum circuit that implements a given sparse state is an important building block that is necessary for many different quantum algorithms. In the context of fault-tolerant quantum computing, the so-called non-Clifford gates are much more expensive to perform than the Clifford ones. We hence provide an algorithm for synthesizing sparse quantum states with a special care for quantum resources. The circuit depth, ancilla count, and crucially non-Clifford count of the circuit produced by the algorithm are all linear in the sparsity when access to arbitrary-angled rotations is given. When compiled down to the standard Clifford+T gate set, several constructions can be given for increasingly better T-count and depth at the expense of a larger number of ancillae. The most optimised construction for T-count reaches $\mathcal O\left(s+\log_2(1/ε)\right)$ T gates for error $ε$ . The constructions are broken into two parts, one that synthesises a generalized W-state, well studied in the literature; and the second which is a classical reversible circuit implementing a permutation that maps the basis states of the W-state to those of the target sparse quantum state. We reduce this problem to the diagonalization of a binary matrix, using a specific set of elementary matrix operations corresponding to the classical reversible gates. We then solve this problem using a new version of Gauss-Jordan elimination, that minimizes the circuit complexities including circuit depth using parallel elimination steps. When the circuit is applied in one direction, we notice that all occurrences of (the expensive) Toffoli gates can all be replaced by adaptive Clifford circuits, leading to a better non-Clifford count.Preparing a quantum circuit that implements a given sparse state is an important building block that is necessary for many different quantum algorithms. In the context of fault-tolerant quantum computing, the so-called non-Clifford gates are much more expensive to perform than the Clifford ones. We hence provide an algorithm for synthesizing sparse quantum states with a special care for quantum resources. The circuit depth, ancilla count, and crucially non-Clifford count of the circuit produced by the algorithm are all linear in the sparsity when access to arbitrary-angled rotations is given. When compiled down to the standard Clifford+T gate set, several constructions can be given for increasingly better T-count and depth at the expense of a larger number of ancillae. The most optimised construction for T-count reaches $\mathcal O\left(s+\log_2(1/ε)\right)$ T gates for error $ε$. The constructions are broken into two parts, one that synthesises a generalized W-state, well studied in the literature; and the second which is a classical reversible circuit implementing a permutation that maps the basis states of the W-state to those of the target sparse quantum state. We reduce this problem to the diagonalization of a binary matrix, using a specific set of elementary matrix operations corresponding to the classical reversible gates. We then solve this problem using a new version of Gauss-Jordan elimination, that minimizes the circuit complexities including circuit depth using parallel elimination steps. When the circuit is applied in one direction, we notice that all occurrences of (the expensive) Toffoli gates can all be replaced by adaptive Clifford circuits, leading to a better non-Clifford count.

arXiv:2508.05386Aug 7, 20250 citations

Quantum Gates and Quantum Algorithms with Clifford Algebra Technique

M. Gregorič, N. S. Mankoč Borštnik

arXiv:0801.3201International Journal of Theoretical PhysicsSep 2, 20086 citations

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

\mathcal{O}(N^m \text{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.

\mathcal O\left(s+\log_2(1/ε)\right)

T gates for error

ε

. The constructions are broken into two parts, one that synthesises a generalized W-state, well studied in the literature; and the second which is a classical reversible circuit implementing a permutation that maps the basis states of the W-state to those of the target sparse quantum state. We reduce this problem to the diagonalization of a binary matrix, using a specific set of elementary matrix operations corresponding to the classical reversible gates. We then solve this problem using a new version of Gauss-Jordan elimination, that minimizes the circuit complexities including circuit depth using parallel elimination steps. When the circuit is applied in one direction, we notice that all occurrences of (the expensive) Toffoli gates can all be replaced by adaptive Clifford circuits, leading to a better non-Clifford count.