Time evolution of quantum gates and the necessity of complex numbers

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 constructreal' 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.

• The paper shows that continuous time evolution of quantum gates inherently produces complex amplitudes, making real-only models inadequate except at isolated instants.
• It employs explicit Hamiltonian modeling to reveal that standard gates and entanglement generation necessitate complex phase evolution, even for systems initially in real states.
• The study concludes that constructing a fully real quantum mechanics either embeds complex structures in higher dimensions or fails to capture essential dynamical and probabilistic features.

Time Evolution of Quantum Gates and the Necessity of Complex Numbers

Introduction and Scope

The paper addresses a foundational issue in quantum theory—the necessity of complex numbers for describing the time evolution of quantum gates and entanglement generation. It systematically examines whether quantum dynamics can be faithfully emulated using only real numbers (rebits and real Hilbert spaces), or if the conventional use of complex numbers is genuinely indispensable. The approach encompasses detailed modeling of quantum logic gates as physical evolutions driven by effective Hamiltonians, analyses of real and complex Hilbert space mappings, and scrutiny of the structural differences between real and complex formulations at both the level of single-qubit and multi-qubit systems.

Temporal Dynamics of Quantum Gates in Complex Hilbert Space

Gate operations in quantum circuits are realized as unitary evolutions, typically modeled as exponentials of Hermitian (Hamiltonian) operators acting for finite time $\tau$. The paper provides explicit constructions of the time-dependent evolution for unary gates such as $Z$, $X$, $Y$, and Hadamard, all defined on $\mathbb{C}^2$. Through spectral decomposition, it is shown that these gates generate continuous trajectories on the Bloch sphere along latitudinal lines associated with their eigenvectors.

A crucial finding is that even for input states restricted to real vectors (rebits), the time evolution under these gates instantly generates complex amplitudes, driving the state off its original real subspace except at discrete instants. Intermediate states invariably exhibit complex components; thus, continuous physical evolution of standard gates cannot be accurately captured within a strictly real formulation. This is illustrated by analyzing the explicit action of $Z(t)$ and related gates on an initial rebit state, with the resulting trajectory necessarily leaving the real axis in Hilbert space except at isolated times.

Entanglement Formation and the Complex Phase

The analysis extends to two-qubit systems, with a focus on the continuous-time generation of entanglement, particularly via the CNOT operation and preparation of Bell states. The interaction Hamiltonian required for entanglement unavoidably involves the complex phase, manifest in non-factorizable phase evolution $e^{-i\omega_{ij}t}$. The construction of canonical entangled states (Bell states) from separable initial conditions is shown to be fundamentally dependent on the complex structure, as phase factors coupling the two subsystems cannot generally be decomposed into real-valued evolutions. Thus, formation of entanglement through physical gate dynamics is inevitably tied to complex amplitudes.

Limitations of Real Hilbert Space Formulations

Addressing real quantum mechanics, the paper demonstrates that the continuous evolution of a quantum state in a real Hilbert space $\mathbb{R}^N$ requires operators in the special orthogonal group $SO(N)$, i.e., evolutions generated by antisymmetric matrices. However, many quantum gates—including the NOT, Hadamard, and CNOT—have determinant $-1$ and hence are not members of $Z$0 or $Z$1, precluding their continuous realization in a strictly real setting of matching dimension.

In particular, the action of these gates cannot be continuously interpolated within $Z$2 (for unary gates) or $Z$3 (for binary gates) using only real orthogonal transformations. The conclusion is that the geometric and algebraic structure required for quantum computation—unitary gates with determinant $Z$4, entanglement generation, and phase coherence—cannot be fully replicated with real-linear algebra on spaces of equivalent dimension.

Complex-to-Real Space Mappings and the Misconception of "Real Quantum Mechanics"

Mappings from $Z$5 to $Z$6 by replacing each complex entry with a $Z$7 real matrix correspond to an isomorphic embedding: all operations preserve the complex algebra, merely represented in a real-valued format. The paper formalizes this as a Kronecker product structure, showing that the resulting real matrices are block representations of their complex counterparts, and any operation so constructed is structurally and functionally equivalent to the original complex operator.

The standard procedure translates unitary operators to special orthogonal ones in the doubled real space, but the complex phase and amplitude information remain encoded in the block structure. Observables—Hermitian matrices—are similarly mapped, with the symmetry properties of real and imaginary components preserved under the mapping. Thus, the "real" formulation presented by this approach fails to eliminate the fundamental role of complex numbers; it only shifts their representation.

Structure of Operators in High-Dimensional Real Vector Spaces

The paper surveys the basis structure of endomorphisms on $Z$8, demonstrating that the subset of antisymmetric operators (those capable of generating $Z$9 evolutions) can each be associated with a complex structure, but not all general symmetric or even antisymmetric operators correspond to complex operations. Mappings from $X$0 to $X$1 do not span the full operator space—only the complex-structured subspace is accessibles. Thus, any "real" quantum mechanics constructed by such mappings remains inherently complex in content, and a strictly real-numbered quantum mechanics would lack the necessary expressive power.

Strong Numerical Results and Theoretical Claims

• The time evolution trajectories generated by standard unary and binary quantum gates necessarily depart from the real subspace at all intermediate times, except at isolated points, precluding strictly real-valued temporal evolution.
• No real orthogonal operator in $X$2 with $X$3 or $X$4 can model the continuous time evolution of standard gates ($X$5, $X$6, $X$7, CNOT), as these gates have determinants $X$8.
• The isomorphic mapping from $X$9 to $Y$0 is merely a complex representation, and does not constitute a genuinely "real" alternative to quantum mechanics.

Practical and Theoretical Implications

These results imply that any attempt to construct a fully real quantum mechanics equivalent to the standard complex formulation either smuggles the complex structure into the formalism via higher-dimensional real representations or fails to capture dynamical and entanglement phenomena central to quantum theory. This has significant consequences for foundational studies—such as recent experimental proposals aiming to falsify real quantum theory formulations [Renou et al., Nature 2021]—and for quantum information architectures that may consider real-valued subtheories (rebits) for comparison or computational subroutines.

From a mathematical physics perspective, this analysis reinforces the strict necessity of complex numbers for both the dynamical and probabilistic consistency of quantum theory as embedded in the Born rule and Schrödinger equation. Any future development of alternative quantum theories must account for the inescapably two-dimensional (modulus and phase) structure encoded in complex amplitudes.

Conclusion

The analysis conclusively demonstrates that the temporally continuous physical evolution required for quantum gate operations—and by extension, quantum computation and entanglement—is indelibly tied to the use of complex numbers. Attempts to recast quantum mechanics in strictly real-valued terms either fail to preserve dynamical and structural features or, in effect, serve only as alternative representations of the original complex theory, merely obfuscating the essential role of the complex phase. Genuine real-numbered quantum mechanics remains insufficient for capturing the inherent dynamics and probabilistic structure demanded by experiment and theoretical consistency.