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All You Need is pi: Quantum Computing with Hermitian Gates

Ben Zindorf, Sougato Bose

TL;DR

This work demonstrates that universal quantum computation can be achieved using solely Hermitian gates, leveraging π-rotations about two fixed axes together with a CNOT. It establishes that any single-qubit operator can be decomposed into two Hermitian π-rotations, and that a minimal Hermitian gate set {CNOT, π-rotations} is universal; restricting to fixed axes yields practical Clifford–like sets. It also introduces amplitude-error suppression via four π-rotations (and two for state preparation), and provides efficient, axis-transformation-based constructions for controlled SU(2)/U(2) and U(4) gates, with favorable CNOT counts in both all-to-all and LNN connectivities. Collectively, these results offer a unified, geometry-driven framework for circuit synthesis and compilation in Hermitian quantum gates, enabling reduced entangling-gate overhead and robust single-qubit operations.

Abstract

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.

All You Need is pi: Quantum Computing with Hermitian Gates

TL;DR

This work demonstrates that universal quantum computation can be achieved using solely Hermitian gates, leveraging π-rotations about two fixed axes together with a CNOT. It establishes that any single-qubit operator can be decomposed into two Hermitian π-rotations, and that a minimal Hermitian gate set {CNOT, π-rotations} is universal; restricting to fixed axes yields practical Clifford–like sets. It also introduces amplitude-error suppression via four π-rotations (and two for state preparation), and provides efficient, axis-transformation-based constructions for controlled SU(2)/U(2) and U(4) gates, with favorable CNOT counts in both all-to-all and LNN connectivities. Collectively, these results offer a unified, geometry-driven framework for circuit synthesis and compilation in Hermitian quantum gates, enabling reduced entangling-gate overhead and robust single-qubit operations.

Abstract

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 rotations up to a global phase. We show that a gate set comprised of rotations about two fixed axes, along with the CNOT gate, is universal for quantum computation. Moreover, we show that two rotations can transform the axis of any multi-controlled unitary, a special case being a single CNOT sufficing for any controlled 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.
Paper Structure (11 sections, 18 theorems, 31 equations, 3 figures, 2 tables)

This paper contains 11 sections, 18 theorems, 31 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Single qubit operators
  3. Hermitian Sets
  4. Amplitude error suppression
  5. Controlled $\Pi$ Gates
  6. Controlled SU(2)/U(2) gates
  7. Controlled U(4) gates
  8. Conclusion
  9. Controlled U(4) gates construction
  10. Proofs of $\Pi$ gate Lemmas
  11. Toffoli decompositions

Key Result

Lemma 1

Any $R_{\hat{v}}(\lambda)\in SU(2)$ operator can be implemented as $\Pi(\hat{v}_2)\Pi(\hat{v}_1)$ with $\hat{v}_1$ as any unit vector perpendicular to $\hat{v}$, and $\hat{v}_2 = \hat{R}_{\hat{v}}(\frac{\lambda}{2})\hat{v}_1$ with $\frac{\lambda}{2}\in (-\pi,\pi]$.

Figures (3)

  • Figure 1: (a) A rotation by $\lambda$ about the vector $\hat{v}$ can be achieved by a $\pi$-rotation about $\hat{v}_1$ followed by a $\pi$-rotation about $\hat{v}_2$. (b) Describing the chosen axes (on the plane perpendicular to $\hat{v}$) for our four $\Pi$ gates decomposition.
  • Figure 2: Numerical gate infidelity (log scale) as a function of the rotation angle $\lambda$ with a fixed axis $\hat{v}(\tfrac{\pi}{2},\tfrac{\pi}{16})$ and an amplitude error $\epsilon=10^{-3}$. Comparing the two $\Pi$, four $\Pi$, Euler ZYZ and the direct $R_v$ implementations.
  • Figure 3: Numerical state infidelity (log scale) as a function of $\lambda$, transforming the $\ket{0}$ state to a final state defined by the state vector $\hat{v}_B=\hat{v}(\lambda,\tfrac{\pi}{2})$. Comparing the single $\Pi$, two $\Pi$, Euler ZYZ and the geodesic ($R_{v_\perp}$) implementations with amplitude error $\epsilon=10^{-3}$.

Theorems & Definitions (31)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 1
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 21 more