arXiv:2607.12907

Unitary Synthesis with Near-Optimal T-Count for Near-Clifford Unitaries

Wang Fang, Chris Heunen, Qisheng Wang

quant-phSubmitted Jul 14, 2026• Announced Jul 15, 2026• 0 citations

Abstract

We present an approach to unitary synthesis that implements an arbitrary $n$-qubit unitary operator $U$ by a Clifford+T circuit with T-count $\widetilde{O}(2^n d_F^{\mathcal{C}}(U))$, where $d_F^{\mathcal{C}}(U)$ is the Frobenius norm distance of $U$ to the Clifford group. The T-count is shown to be near-optimal when $d_F^{\mathcal{C}}(U)$ is a constant. Our approach improves the previous best upper bound $\widetilde{O}(2^{4n/3})$ due to Tan (2025) for a large class of unitary operators $U$ as long as $d_F^{\mathcal{C}}(U) \ll 2^{n/3}$.

Comments: 24 pages

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arXiv:2607.12907

Unitary Synthesis with Near-Optimal T-Count for Near-Clifford Unitaries

Wang Fang, Chris Heunen, Qisheng Wang

quant-phSubmitted Jul 14, 2026• Announced Jul 15, 2026• 0 citations

arXiv HTML PDF

Abstract

We present an approach to unitary synthesis that implements an arbitrary

n

-qubit unitary operator

U

by a Clifford+T circuit with T-count

\widetilde{O}(2^n d_F^{\mathcal{C}}(U))

, where

d_F^{\mathcal{C}}(U)

is the Frobenius norm distance of

U

to the Clifford group. The T-count is shown to be near-optimal when

d_F^{\mathcal{C}}(U)

is a constant. Our approach improves the previous best upper bound

\widetilde{O}(2^{4n/3})

due to Tan (2025) for a large class of unitary operators

U

as long as

d_F^{\mathcal{C}}(U) \ll 2^{n/3}

Comments: 24 pages

Related papers

$T$ -efficient unitary synthesis via block-sparse Hamiltonian simulation$T$-efficient unitary synthesis via block-sparse Hamiltonian simulation

Diyi Liu, Shuchen Zhu, Wibe Albert de Jong +3 more

We construct Clifford+ $T$ circuits implementing arbitrary $n$ -qubit unitaries with worst-case $T$ -count $\widetilde{O}(N^{8/7})$ at constant operator-norm error, where $N=2^n$ , improving the previous exponent $5/4$ . The construction uses $\widetilde{O}(N^{6/7})$ clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error $ε\in(0,1/2)$ , let $L=\log_2(N/ε)$ , the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using $\widetilde{O}(N^{8/7}L^{3/7}+NL)$ $T$ gates and $\widetilde{O}(N)$ clean ancillas whenever $L\le N$ . We also prove that, at constant precision, the exponent $8/7$ is optimal within a sequential block-SELECT model.We construct Clifford+$T$ circuits implementing arbitrary $n$-qubit unitaries with worst-case $T$-count $\widetilde{O}(N^{8/7})$ at constant operator-norm error, where $N=2^n$, improving the previous exponent $5/4$. The construction uses $\widetilde{O}(N^{6/7})$ clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error $ε\in(0,1/2)$, let $L=\log_2(N/ε)$, the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using $\widetilde{O}(N^{8/7}L^{3/7}+NL)$ $T$ gates and $\widetilde{O}(N)$ clean ancillas whenever $L\le N$. We also prove that, at constant precision, the exponent $8/7$ is optimal within a sequential block-SELECT model.

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Mathias Weiden, Jae Won Kim, Justin Kalloor +2 more

For the standard Clifford+ $T$ gate set, deterministic, ancilla-free synthesis now attains the minimal $T$ -count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The $\sqrt{T}$ gate rotates by half the angle of $T$ , 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+ $\sqrt{T}$ 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 $\varepsilon=10^{-3}$ to $10^{-8}$ , the cost of Clifford+ $\sqrt{T}$ circuits scales as $2.4\log_2(1/\varepsilon)$ compared to $3.0\log_2(1/\varepsilon)$ for $T$ -count-optimal Clifford+ $T$ circuits. Once a one-time catalyst state is amortized, the Clifford+ $\sqrt{T}$ circuits are never costlier than their Clifford+ $T$ counterparts.For the standard Clifford+$T$ gate set, deterministic, ancilla-free synthesis now attains the minimal $T$-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The $\sqrt{T}$ gate rotates by half the angle of $T$, 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+$\sqrt{T}$ 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 $\varepsilon=10^{-3}$ to $10^{-8}$, the cost of Clifford+$\sqrt{T}$ circuits scales as $2.4\log_2(1/\varepsilon)$ compared to $3.0\log_2(1/\varepsilon)$ for $T$-count-optimal Clifford+$T$ circuits. Once a one-time catalyst state is amortized, the Clifford+$\sqrt{T}$ circuits are never costlier than their Clifford+$T$ counterparts.

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(Almost) quadruply optimal unitary designs in 1D

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We construct $n$ -qubit approximate unitary $k$ -designs in 1D systems, achieving circuit depth $O(\log(n/\varepsilon) + k\log k)$ with relative error $\varepsilon$ and requiring $O(nk\log k)$ magic gates. This matches existing lower bounds $Ω(\log(n/\varepsilon) + k)$ for circuit depth, and $\widetildeΩ(nk)$ for the required number of $T$ gates, up to a $\log k$ factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from $O(k\log k)$ to $O(\log k)$ . We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making $O(\log k)$ -local random unitaries realizable in depth $O(k\log k)$ . As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.We construct $n$-qubit approximate unitary $k$-designs in 1D systems, achieving circuit depth $O(\log(n/\varepsilon) + k\log k)$ with relative error $\varepsilon$ and requiring $O(nk\log k)$ magic gates. This matches existing lower bounds $Ω(\log(n/\varepsilon) + k)$ for circuit depth, and $\widetildeΩ(nk)$ for the required number of $T$ gates, up to a $\log k$ factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from $O(k\log k)$ to $O(\log k)$. We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making $O(\log k)$-local random unitaries realizable in depth $O(k\log k)$. As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.

quant-pharXiv:2608.18650Aug 20, 20260 citations

CS-count-optimal quantum circuits for arbitrary multi-qubit unitaries

Priyanka Mukhopadhyay

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 ${\mathcal {J}}_n^{CS}\subset {\mathcal {J}}_n^T$ J n CS ⊂ J n T , where ${\mathcal {J}}_n^{CS}$ J n CS and ${\mathcal {J}}_n^T$ 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 ${\mathcal {J}}_n^{Tof}={\mathcal {J}}_n^{CS}$ J n Tof = J n CS , where ${\mathcal {J}}_n^{Tof}$ 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.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 $${\mathcal {J}}_n^{CS}\subset {\mathcal {J}}_n^T$$ J n CS ⊂ J n T , where $${\mathcal {J}}_n^{CS}$$ J n CS and $${\mathcal {J}}_n^T$$ 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 $${\mathcal {J}}_n^{Tof}={\mathcal {J}}_n^{CS}$$ J n Tof = J n CS , where $${\mathcal {J}}_n^{Tof}$$ 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.

Scientific ReportsJun 17, 20242 citations

We present an approach to unitary synthesis that implements an arbitrary

n

-qubit unitary operator

U

by a Clifford+T circuit with T-count

\widetilde{O}(2^n d_F^{\mathcal{C}}(U))

, where

d_F^{\mathcal{C}}(U)

is the Frobenius norm distance of

U

to the Clifford group. The T-count is shown to be near-optimal when

d_F^{\mathcal{C}}(U)

is a constant. Our approach improves the previous best upper bound

\widetilde{O}(2^{4n/3})

due to Tan (2025) for a large class of unitary operators

U

as long as

d_F^{\mathcal{C}}(U) \ll 2^{n/3}

T

-efficient unitary synthesis via block-sparse Hamiltonian simulation

We construct Clifford+

T

circuits implementing arbitrary

n

-qubit unitaries with worst-case

T

-count

\widetilde{O}(N^{8/7})

at constant operator-norm error, where

N=2^n

, improving the previous exponent

5/4

. The construction uses

\widetilde{O}(N^{6/7})

clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error

ε\in(0,1/2)

, let

L=\log_2(N/ε)

, the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using

\widetilde{O}(N^{8/7}L^{3/7}+NL)

T

gates and

\widetilde{O}(N)

clean ancillas whenever

L\le N

. We also prove that, at constant precision, the exponent

8/7

is optimal within a sequential block-SELECT model.

Synthesizing arbitrary

n

-qubit unitaries using as few non-Clifford gates as possible is a central problem in fault-tolerant quantum compilation. We present a Clifford+

T

quantum circuit construction that approximately implements any classically specified unitary to within error

ε

and achieves a worst-case

T

-count with leading exponential scaling of

2^{5n/4}

whenever

\log(1/ε)=\operatorname{poly}(n)

. This improves upon the best previous

2^{4n/3}

scaling. The key innovation lies in treating the target unitary as a single block-encoded object rather than a long product of simpler operations. A technique of block flattening controls the normalization while preserving an efficient implementation of the block encoding; subsequently, quantum singular value transformation maps its common singular value to one, thereby recovering the target unitary.

Approximate synthesis of general single-qubit unitaries over the Clifford+

\sqrt{T}

gate set

For the standard Clifford+

T

gate set, deterministic, ancilla-free synthesis now attains the minimal

T

-count for general single-qubit unitaries (Morisaki et al., arXiv:2510.05816). The

\sqrt{T}

gate rotates by half the angle of

T

, 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+

\sqrt{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

\varepsilon=10^{-3}

10^{-8}

, the cost of Clifford+

\sqrt{T}

circuits scales as

2.4\log_2(1/\varepsilon)

compared to

3.0\log_2(1/\varepsilon)

for

T

-count-optimal Clifford+

T

circuits. Once a one-time catalyst state is amortized, the Clifford+

\sqrt{T}

circuits are never costlier than their Clifford+

T

counterparts.

We construct

n

-qubit approximate unitary

k

-designs in 1D systems, achieving circuit depth

O(\log(n/\varepsilon) + k\log k)

with relative error

\varepsilon

and requiring

O(nk\log k)

magic gates. This matches existing lower bounds

Ω(\log(n/\varepsilon) + k)

for circuit depth, and

\widetildeΩ(nk)

for the required number of

T

gates, up to a

\log k

factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from

O(k\log k)

O(\log k)

. We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making

O(\log k)

-local random unitaries realizable in depth

O(k\log k)

. As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.

{\mathcal {J}}_n^{CS}\subset {\mathcal {J}}_n^T

J n CS ⊂ J n T , where

{\mathcal {J}}_n^{CS}

J n CS and

{\mathcal {J}}_n^T

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

{\mathcal {J}}_n^{Tof}={\mathcal {J}}_n^{CS}

J n Tof = J n CS , where

{\mathcal {J}}_n^{Tof}

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.