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arXiv:2603.13169 · Journal of the Physical Society of Japan

Circuit Optimization for Universality Transformation

Yasuaki Nakayama, Yuki Takeuchi, Seiseki Akibue

quant-phSubmitted Mar 13, 2026• Announced Mar 16, 2026• 0 citations• Revised Oct 2, 2026 (v3)
v1 · Mar 13, 2026v2 · Mar 29, 2026v3 · Oct 2, 2026
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Abstract

It is known that a computationally universal gate set $\{H,CCZ\}$ can be transformed to a strictly universal one $\{H, Λ(S)\}$ using one maximally imaginary state $|+i \rangle$ and non-imaginary ancillary qubits. We succeed this transformation with a shorter circuit that eliminates non-imaginary ancillary qubits. We further extend this to the continuous gate-set setting, showing that any multi-qubit unitary can be exactly generated by real single-qubit unitary gates, $CCZ$ gates and $|0 \rangle |+i \rangle$.

Journal reference
J. Phys. Soc. Jpn. 95, 085001 (2026)
DOI
10.7566/JPSJ.95.085001
Comments
2 pages, 1 figure. Revised in response to referee comments

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It is known that a computationally universal gate set {H,CCZ}\{H,CCZ\}{H,CCZ} can be transformed to a strictly universal one {H,Λ(S)}\{H, Λ(S)\}{H,Λ(S)} using one maximally imaginary state ∣+i⟩|+i \rangle∣+i⟩ and non-imaginary ancillary qubits. We succeed this transformation with a shorter circuit that eliminates non-imaginary ancillary qubits. We further extend this to the continuous gate-set setting, showing that any multi-qubit unitary can be exactly generated by real single-qubit unitary gates, CCZCCZCCZ gates and ∣0⟩∣+i⟩|0 \rangle |+i \rangle∣0⟩∣+i⟩.
Unitary synthesis underlies universal quantum computation by translating unitary transformations into circuits of elementary gates. We show that for any budget m≥0m\geq 0m≥0 of ancilla, every nnn-qubit unitary can be exactly implemented using a quantum circuit consisting of O(4n)O(4^n)O(4n) CNOT and single-qubit gates in O(n+4n/(n+m))O(n+4^n/(n+m))O(n+4n/(n+m)) layers attaining the asymptotically optimal depth--ancilla tradeoff while simultaneously achieving optimal circuit size. The key idea is to encode computational basis states as wavefunctions sharing a Gaussian envelope, converting unitary action on amplitudes into coordinate transformations that admit shallow circuit implementations. The continuous encoding is then discretized, with approximation errors removed by exact residual correction. The depth-ancilla tradeoff follows by recursively applying the quantum Shannon decomposition, synthesizing its smaller unitary factors, and reusing ancillary qubits.
The computational universality with an elementary gate set {H,CCZ}\{H,CCZ\}{H,CCZ} can be transformed to the strict universality by using a maximally imaginary state ∣+i⟩|+i \rangle∣+i⟩ and some non-imaginary ancillary qubits. From the viewpoint of operational resource theory, it would be intriguing to elucidate a resource for the universality transformation. In this paper, we consider the exact universality transformation in which arbitrary real orthogonal matrices can be applied and a supplied resource state is used to simulate unitary operations exactly and deterministically. Within this operational model, we explore a necessary and sufficient condition for resource states to realize the universality transformation under free real operations. We show that ∣+i⟩|+i \rangle∣+i⟩ is a unique resource state up to the free operations. Moreover, we obtain a stronger conclusion. If a given resource state cannot be used for the universality transformation, then realizable quantum gates are restricted to unitary matrices proportional to real orthogonal matrices. Therefore, we can tell that ∣+i⟩|+i \rangle∣+i⟩ is unique (up to the free operations) not only as a state whose resource measure of imaginarity is maximal, but also as a state which empowers real operations with the ability to apply at least one non-real quantum gate (regardless of the magnitudes of its imaginary parts).
A 1-qubit gate is defined as an element of the unitary group U(2)U(2)U(2), the group of complex valued matrices whose conjugate transposes equal their inverses. In quantum computing, it is important to be able to find good approximations of 1-qubit gates by using a dense group generated by a universal subset of the projective special unitary group G=PSU(2)G=PSU(2)G=PSU(2). Here, PSU(2)PSU(2)PSU(2) is the group formed by taking all elements of U(2)U(2)U(2) with determinant 1 and taking the quotient of this subgroup by the group {I,−I}\{I,-I\}{I,−I} where III is the identity matrix. In this http URL [1], Sarnak recently studied this problem motivated by developing a continued fraction type algorithm for this problem. The continued fraction algorithm is the fastest known algorithm to approximate real numbers by the dense subset of rational numbers. In this report, we study the efficiency of the approximation of GGG by using a dense group generated by a universal subset of GGG. The measure of the efficiency K(g)K(g)K(g) of an arbitrary universal subset ggg is defined in [1] and satisfies K(g)⩾1K(g)\geqslant 1K(g)⩾1. ggg is more efficient when K(g)K(g)K(g) is closer to 111. We address some open problems posed by Sarnak in [1]. This report forms the basis of the shorter preprint [LD].
Universal gate sets for quantum computation, when single and two qubit operations are accessible, include both Hermitian and non-Hermitian gates. Here we utilize the fact that any single-qubit operator may be implemented as two Hermitian gates, and thus a purely Hermitian universal set is possible. This implementation can be used to prepare high fidelity single-qubit states in the presence of amplitude errors, and helps to achieve a high fidelity single-qubit gate decomposition using four Hermitian gates. An implementational convenience can be that non-identity single-qubit Hermitian gates are equivalent to π rotations up to a global phase. We show that a gate set comprised of π rotations about two fixed axes, along with the CNOT gate, is universal for quantum computation. Moreover, we show that two π rotations can transform the axis of any multi-controlled unitary, a special case being a single CNOT sufficing for any controlled π rotation. These gates simplify the process of circuit compilation in view of their Hermitian nature. We exemplify by designing efficient circuits for a variety of controlled gates, and achieving a CNOT count reduction for the four-controlled Toffoli gate in LNN-restricted qubit connectivity.
We find the optimal universal way of manipulating a single qubit, $|\ensuremath{\psi}(\ensuremath{\vartheta},\ensuremath{\varphi})〉,$ such that $(\ensuremath{\vartheta},\ensuremath{\varphi})\ensuremath{\rightarrow}(\ensuremath{\vartheta}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\varphi}\ensuremath{-}\ensuremath{\beta}).$ Such optimal transformations fall into two classes. For $0<~\ensuremath{\alpha}<~\ensuremath{\pi}/2,$ the optimal map is the identity and the fidelity varies monotonically from 1 (for $\ensuremath{\alpha}=0)$ to 12\frac{1}{2}21​ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2).$ For $\ensuremath{\pi}/2<~\ensuremath{\alpha}<~\ensuremath{\pi},$ the optimal map is the universal-NOT gate and the fidelity varies monotonically from 12\frac{1}{2}21​ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2)$ to 23\frac{2}{3}32​ (for $\ensuremath{\alpha}=\ensuremath{\pi}).$ The fidelity 23\frac{2}{3}32​ is equal to the fidelity of measurement. It is therefore rather surprising that for some values of $\ensuremath{\alpha}$ the fidelity is lower than 23.\frac{2}{3}.32​. For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.