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arXiv:2605.28152

Non-Hermitian Computers Need No Complex Numbers

Qi Zhang

quant-phSubmitted May 27, 2026• Announced May 28, 2026• 0 citations
arXivHTMLPDF

Abstract

In traditional quantum computing, it has been established that real quantum computation augmented with non-Clifford gates is as powerful as universal quantum computation. Here we investigate this phenomenon in the non-Hermitian setting. We show that a non-Hermitian quantum computer equipped with the real gate set ${H, \text{CCNOT}, G}$, where $G = \operatorname{diag}(g^{-1}, g)$ with $g > 0$ and $g \neq 1$, can solve problems in $\text{P}^{\sharp\text{P}}$ in polynomial time, matching the capability of its universal non-Hermitian counterpart ${H, T, \text{CNOT}, G}$. This demonstrates that non-unitarity, rather than universality, is the essential resource, and that complex numbers are unnecessary.

Comments
10 pages, 4 figures

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In traditional quantum computing, it has been established that real quantum computation augmented with non-Clifford gates is as powerful as universal quantum computation. Here we investigate this phenomenon in the non-Hermitian setting. We show that a non-Hermitian quantum computer equipped with the real gate set H,CCNOT,G{H, \text{CCNOT}, G}H,CCNOT,G, where G=diag⁡(g−1,g)G = \operatorname{diag}(g^{-1}, g)G=diag(g−1,g) with g>0g > 0g>0 and g≠1g \neq 1g=1, can solve problems in P♯P\text{P}^{\sharp\text{P}}P♯P in polynomial time, matching the capability of its universal non-Hermitian counterpart H,T,CNOT,G{H, T, \text{CNOT}, G}H,T,CNOT,G. This demonstrates that non-unitarity, rather than universality, is the essential resource, and that complex numbers are unnecessary.
We elucidate the profound connection between physics and computation by proposing and examining the model of the non-Hermitian quantum computer (NQC). In addition to conventional quantum gates such as the Hadamard, phase, and CNOT gates, this model incorporates a non-unitary quantum gate GGG. We show that NQC is extraordinarily powerful, capable of solving not only all NP problems but also all problems within the complexity class P♯P\text{P}^{\sharp\text{P}}P♯P in polynomial time. We investigate two physical schemes for implementing the non-unitary gate GGG and find that the remarkable computational power of NQC originates from the exponentially large physical resources required.
Non-Hermitian (NH) quantum systems demonstrate striking differences from their Hermitian counterparts, leading to claims of NH advantage in areas ranging from metrology to entanglement generation. We show that in the context of quantum computation, any such NH advantage is unlikely to be scalable as an efficient computational resource: If coherent nonunitary evolution with renormalization could be realized with only polynomial overhead, then the resulting model could implement postselection, implying implausibly strong complexity-theoretic power under standard assumptions. We define NHBQP ( U ) as the computational power of polynomial-size quantum circuits that, in addition to a standard universal unitary gate set, may apply a fixed gate U on O ( 1 ) qubits that is not proportional to a unitary, with the state renormalized after each use of U . We prove that this model is powerful enough to decide every language in PostBQP (equivalently PP ). Moreover, in the standard uniform circuit-family model, this characterization is tight: For any fixed such nonunitary gate U , NHBQP ( U ) = PostBQP = PP . PostBQP is believed intractable, so this suggests that any scalable NH computational advantage must come with a compensating cost limiting its efficiency. Additionally, we study simulation complexity of restricted classes of nonunitary systems by purifying them to postselected unitary evolution in a form preserving system locality. Using this framework, we show that unitary gates with postselection can simulate not only evolution under NH Hamiltonians but arbitrary quantum trajectories. Any NH model whose purification lies in a strongly simulable unitary family (e.g., Clifford, matchgate, or low-bond-dimension tensor-network circuits) remains efficiently classically simulable, provided the relevant postselected events occur with probability Ω ( 2 − poly ( n ) ) . Thus, adding non-Hermiticity to a universal unitary system makes it infeasibly computationally powerful, while adding it to a strongly simulable system adds no computational power in this setting.
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.
Non-Hermitian systems offer new opportunities for quantum optimization and computation. Here, we show that non-Hermitian quantum adiabatic algorithms require not only a real, gapped spectrum, but also a stable pseudospectrum. We propose a novel framework by mapping non-unitary quantum circuits to local Hamiltonian paths, thereby preserving their optimization advantages and shallow depth. While a direct non-Hermitian extension of the Feynman-Kitaev construction suffers severe pseudospectral instability, our history-decoupled construction yields both a controlled pseudospectrum and a real, gapped spectrum. Using the CK benchmark family of maximum independent set problems, we demonstrate polynomial-evolution-time non-Hermitian adiabatic computation that remains robust against perturbations. We further discuss a feasible optical implementation using coupled waveguides with an auxiliary lossy channel. Our work establishes pseudospectral stability, alongside real, gapped spectra, as key principles and a practical route for non-Hermitian quantum adiabatic computation.
We study the resources required to achieve universal quantum computing via the gate sets that provide the fundamental instructions from which quantum algorithms are built. While single-gate universal sets are known, they rely on precisely tuned irrational rotations, making them difficult to realize on near-term devices. We find that the controlled-VVV gate, an elementary two-qubit interaction directly implementable on leading hardware, is universal and capable of simulating standard universal gate sets with minimal overhead. Specifically, we use catalytic embeddings to develop a constant-overhead algorithm that simulates standard universal gate sets, including Clifford+T+T+T and Clifford+++Toffoli. We combine this simulation algorithm with existing synthesis results to yield exact and approximate synthesis algorithms for unitaries with and without number-theoretic restrictions. The results highlight how full quantum computational power, complete with algorithms for synthesis and simulation, can emerge from unexpectedly simple ingredients.
Abstract A universal set of gates for (classical or quantum) computation is a set of gates that can be used to approximate any other operation. It is well known that a universal set for classical computation augmented with the Hadamard gate results in universal quantum computing. Motivated by the latter, we pose the following question: can one perform universal quantum computation by supplementing a set of classical gates with a quantum control, and a set of quantum gates operating solely on the latter? In this work we provide an affirmative answer to this question by considering a computational model that consists of 2n target bits together with a set of classical gates controlled by log (2n + 1) ancillary qubits. We show that this model is equivalent to a quantum computer operating on n qubits. Furthermore, we show that even a primitive computer that is capable of implementing only SWAP gates, can be lifted to universal quantum computing, if aided with an appropriate quantum control of logarithmic size. Our results thus exemplify the information processing power brought forth by the quantum control system.