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arXiv:2604.17086

Time evolution of quantum gates and the necessity of complex numbers

M. P. Vaughan

quant-phcond-mat.mtrl-sciSubmitted Apr 18, 2026• Announced Apr 21, 2026• 0 citations
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Abstract

As physical systems, qubits must evolve from input to output state. We describe a simple scheme in which the effect of a quantum gate is described by the action of an effective Hamiltonian acting for some characteristic time. This model shows that the action of common unary gates is to induce Bloch sphere trajectories along lines of latitude relative to an eigenvector of the gate. Such trajectories would immediately move a `rebit', initially confined to a line of latitude, off this line and acquire a complex phase. The role of the complex phase in bringing about the entanglement of two qubits is also highlighted. It is then asked whether such dynamics could be modelled using real QM. It is shown that the continuous evolution required for such dynamics can only be provided by members of the special orthogonal group of the vector space. Since the matrices representing many quantum gates of interest have determinant -1, no real special orthogonal operators can model their evolution if the dimension of the real vector space is the same as that of the complex space. Next we look at the mapping from a complex vector space of dimension $N$ to a real space of dimension $2N$ that is often used to construct `real' QM. It is shown that this is just an isomorphic mapping from the scalar representation of complex numbers to their $2\times2$ matrix equivalents, so that the resulting matrices are actually represent complex matrices. Finally, we investigate the endomorphism of real vector spaces of dimension $N = 2^{n}$, where $n \in \mathbb{Z}^{+}$, suitable for the modelling of $n-1$ qubits. We confirm that the mapping from $\mathbb{C}^{2^{n-1}} \to \mathbb{R}^{2^{n}}$ only maps elements of $\mathrm{End}(\mathbb{C}^{2^{n-1}})$ to a restricted subspace of $\mathrm{End}(\mathbb{R}^{2^{n}})$ that reproduces the `real' representation of complex matrices.

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Unitary States and Gates of Three-Qubit Quantum Systems

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As physical systems, qubits must evolve from input to output state. We describe a simple scheme in which the effect of a quantum gate is described by the action of an effective Hamiltonian acting for some characteristic time. This model shows that the action of common unary gates is to induce Bloch sphere trajectories along lines of latitude relative to an eigenvector of the gate. Such trajectories would immediately move a rebit', initially confined to a line of latitude, off this line and acquire a complex phase. The role of the complex phase in bringing about the entanglement of two qubits is also highlighted. It is then asked whether such dynamics could be modelled using real QM. It is shown that the continuous evolution required for such dynamics can only be provided by members of the special orthogonal group of the vector space. Since the matrices representing many quantum gates of interest have determinant -1, no real special orthogonal operators can model their evolution if the dimension of the real vector space is the same as that of the complex space. Next we look at the mapping from a complex vector space of dimension $N$ to a real space of dimension $2N$ that is often used to construct real' QM. It is shown that this is just an isomorphic mapping from the scalar representation of complex numbers to their 2×22\times22×2 matrix equivalents, so that the resulting matrices are actually represent complex matrices. Finally, we investigate the endomorphism of real vector spaces of dimension N=2nN = 2^{n}N=2n, where n∈Z+n \in \mathbb{Z}^{+}n∈Z+, suitable for the modelling of n−1n-1n−1 qubits. We confirm that the mapping from C2n−1→R2n\mathbb{C}^{2^{n-1}} \to \mathbb{R}^{2^{n}}C2n−1→R2n only maps elements of End(C2n−1)\mathrm{End}(\mathbb{C}^{2^{n-1}})End(C2n−1) to a restricted subspace of End(R2n)\mathrm{End}(\mathbb{R}^{2^{n}})End(R2n) that reproduces the `real' representation of complex matrices.
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Pure states of two-qubit quantum systems are described by a four-dimensional vector of complex numbers, and unitary operators transferring a two-qubit quantum system from one state to another have the form of a 4 × 4 matrix with complex elements. This fact brings to mind the idea of studying the spaces of four-dimensional numbers with complex components. Moreover, the results obtained by the authors for four-dimensional numbers with real components inspire some optimism. In this paper we construct four-dimensional spaces of complex numbers by analogy with four-dimensional spaces of real numbers. Each four-dimensional number is mapped to a matrix formed from its components and it is proved that the constructed mapping is a bijection and a homomorphism. In the space of fourdimensional numbers, of the eight basis elements, half are real and half are imaginary. The presence of such symmetry distinguish these spaces from the space of quaternions, in which one basis element is real and the rest are imaginary. The symmetry of the basis numbers makes these spaces a natural generalization of one-dimensional and two-dimensional (complex) algebra. The conditions under which the corresponding matrices are gates for two-qubit quantum systems are defined. The notion of unitary state of a two-qubit quantum system is introduced, to which various gates from commutative groups of gates correspond. It is shown that any gate of a unitary state transforms a unitary state into a unitary state and a non-unitary state into a non-unitary state. Almost all gates used in the construction of quantum circuits, in particular H, SWAP, CX, CY, CZ, have the same properties. The problem of searching for a gate that transfers a quantum system from one unitary state to another unitary state has been solved. Thus, with the help of fourdimensional spaces of complex numbers it was possible to construct whole classes of two-qubit gates, which opens new possibilities for the construction of quantum algorithms. The results obtained have important theoretical and practical implications for quantum computing.
This work introduces a complete and self-consistent framework for quantum computation within the domain of real-valued quantum mechanics. Standard quantum computation is predicated on the manipulation of states in a complex Hilbert space, where a complex phase is a critical resource. We demonstrate that this complex structure is not fundamental but can be fully reproduced by a real-operator algebra. Our formalism is built upon a rigorous algebraic isomorphism between the imaginary unit i and the temporal Hilbert transform operator Ht\mathcal{H}_tHt​, which acts as a physical generator of phase. Leveraging this isomorphism, we systematically redefine the fundamental quantum logic gates-including the Pauli, Hadamard, phase rotation, and CNOT gates-entirely within a real operator algebra. We prove that each gate's action on a real-valued qubit state representation is mathematically and operationally equivalent to its conventional complex counterpart. The Pauli-Y, phase, and controlled-phase gates, which are intrinsically complex in the standard model, are shown to correspond to real-operator matrices involving the Hilbert transform. This formulation provides a physically grounded interpretation of quantum logic, where abstract phase manipulation corresponds to the application of a concrete, albeit non-local, integral transform. We demonstrate how this framework correctly reproduces the generation of quantum entanglement and establishes a consistent measurement theory via an Analytic Born Rule in Unified Quantum Mechanics. This work solidifies the viability of a complete real-valued model for quantum information processing and suggests new paradigms for the physical implementation of quantum hardware.
It is argued that many real-valued constructions of quantum mechanics are only \emph{nominally} real in that the operators and states are restricted to impose a \emph{complex structure} on the Hilbert space. That is, complex algebra between pairs of elements representing complex numbers is preserved in these formulations. It is therefore mistaken to think of these constructions as being `real' as they are actually a representation of complex linear algebra. It is then shown that under the assumptions of state normalisation and strict energy conservation, this complex structure (at the very least) is required to allow for time evolution in quantum mechanics. This is only a \emph{minimum} requirement, as our arguments do not preclude the formulation of quantum mechanics in terms of hyper-complex entities, such as quaternions.
The necessity of complex numbers in quantum mechanics has long been debated. This paper develops a real Kahler space formulation of quantum mechanics [19], asserting equivalence to the standard complex Hilbert space framework. By mapping the complex Hilbert space Cn{\mathbb C}^nCn to a real Kahler space K2nK ^{2n}K2n, i.e. R2n{\mathbb R}^{2n}R2n equipped with a metric, a symplectic structure and an automorphism, we establish a correspondence between Hermitian operators in Cn{\mathbb C}^nCn and real operators in K2nK^{2n}K2n. While the isomorphism appears straightforward some subtleties emerge: (i) the overcounting of composite system states under tensor products in R2n{\mathbb R}^{2n}R2n, and (ii) the double degeneracy of operator spectra in the real formulation. Through a systematic investigation of these challenges, we clarify the structural relationship between real and complex formulations, resolve ambiguities in composite system representations, and analyze spectral consequences. Our results reaffirm the equivalence of the two frameworks while highlighting nuanced distinctions with implications for foundational debates on locality, phase invariance, and the role of complex numbers in quantum theory.
Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by 8×88 \times 88×8 complex matrices. Motivated by this correspondence, we extend our algebraic model for two-qubit systems to eight-dimensional spaces endowed with commutative and associative multiplication. To each eight-dimensional number we assign a matrix and prove that the resulting correspondence is a homomorphic bijection. We derive conditions under which the associated matrix is unitary and hence represents a three-qubit quantum gate. This leads to a specific algebraic notion of a unitary state: a normalized state whose multiplication matrix is unitary. We construct 16,384 basis spaces; within each fixed space, multiplication generates a commuting family of quantum gates. For any two unitary states in the class considered, we give an explicit transition-gate construction within one of these algebraic families. For real components, the unitary states in a representative basis space form a union of thirty-two four-dimensional manifolds in R8\mathbb{R}^{8}R8. The construction therefore yields new structured classes of three-qubit unitary operators and invariant state sets, while its extension to larger multiqubit systems remains open.