Time evolution of quantum gates and the necessity of complex numbers

M.P. Vaughan

arxiv: 2604.17086 · v1 · submitted 2026-04-18 · 🪐 quant-ph · cond-mat.mtrl-sci

Time evolution of quantum gates and the necessity of complex numbers

M.P. Vaughan This is my paper

Pith reviewed 2026-05-10 06:26 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mtrl-sci

keywords quantum gatestime evolutioncomplex numbersreal quantum mechanicsBloch spheredeterminantspecial orthogonal groupeffective Hamiltonian

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The pith

Quantum gates with determinant -1 cannot evolve continuously under real special orthogonal transformations of matching dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models quantum gate action as the continuous time evolution produced by a single effective Hamiltonian. This evolution traces lines of latitude on the Bloch sphere relative to an eigenvector of the gate and necessarily introduces complex phases. A real qubit confined to a line of latitude would be displaced by such motion. The author shows that continuous evolution is possible only through special orthogonal matrices, which always have determinant +1. Many standard quantum gates are represented by matrices with determinant -1, so no real orthogonal operator of the same dimension can generate the required trajectories.

Core claim

Because the matrices for common quantum gates have determinant -1, the continuous time evolution they induce cannot be realized by any member of the special orthogonal group when the real vector space has the same dimension as the complex space. Mappings that embed a complex N-dimensional space into a real 2N-dimensional space are isomorphic representations that still carry the complex structure, and higher-dimensional real spaces for multiple qubits only reproduce a restricted subspace equivalent to the complex case.

What carries the argument

The special orthogonal group SO(n) of continuous rotations from the identity, contrasted with the full orthogonal group O(n) that includes determinant -1 elements, applied to the effective-Hamiltonian trajectories on the Bloch sphere.

If this is right

Real-number quantum mechanics cannot generate the continuous dynamics of standard unary gates without leaving the allowed real subspace.
Complex phases acquired during gate evolution are required to produce the entanglement observed between two qubits.
Embeddings of complex quantum mechanics into larger real vector spaces remain equivalent to complex representations and do not remove the need for complex structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The necessity of complex numbers appears at the level of dynamics rather than static states alone.
Discrete gate applications without requiring continuous time evolution might evade the determinant obstruction in a purely real model.

Load-bearing premise

The action of a quantum gate can be accurately modeled as continuous time evolution generated by a single effective Hamiltonian that produces trajectories along lines of latitude on the Bloch sphere.

What would settle it

An explicit construction of a real special orthogonal matrix sequence that reproduces the time evolution of a gate whose complex matrix has determinant -1, while keeping the state space dimension equal to the complex case.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims real QM cannot handle common gates because their det=-1 matrices lie outside SO(N) in equal dimension, using effective-Hamiltonian latitude trajectories on the Bloch sphere, but this rests on an idealized single-H model.

read the letter

The core claim is that modeling a gate as exp(-i H t) for fixed Hermitian H moves a real state vector off its latitude line relative to an eigenvector, forcing a complex phase, and that SO(N) cannot reach the det=-1 matrices of gates like NOT when dimensions match. Vaughan then shows the standard C^N to R^{2N} map is just the 2x2 matrix representation of complex scalars, and that the endomorphism map from C^{2^{n-1}} to R^{2^n} only reproduces the complex subspace of End(R^{2^n}). This is the new angle: linking the specific Bloch trajectories to the group obstruction and checking the multi-qubit isomorphism explicitly. The derivations are clean and use standard facts about unitary groups and orthogonal representations without free parameters or circularity. The latitude picture and the phase-entanglement link are presented plainly. The main soft spot is the single effective Hamiltonian assumption. Real gates are built from time-dependent controls, composite pulses, or adiabatic paths whose net unitary need not follow a one-parameter subgroup, so a real dynamics could in principle reach the same map without the described phase acquisition. The paper does not demonstrate that every gate must be realizable this way or rule out alternative continuous paths in a real setting. The multi-qubit endomorphism claim is sketched but would benefit from explicit matrix checks for n>2. This is for readers already interested in real versus complex formulations of quantum mechanics or attempts to build real-valued quantum information. It is not a broad foundational overhaul but a focused group-theoretic observation on gates. A serious referee should see it because the math is straightforward, the claim is specific enough to test, and the question of whether real QM can reproduce gate sets remains open even if this particular route has a gap.

Referee Report

2 major / 0 minor

Summary. The paper claims that quantum gates can be modeled as the continuous action of a fixed effective Hamiltonian over a characteristic time, producing Bloch-sphere trajectories along lines of latitude relative to an eigenvector. This immediately forces a complex phase for any real initial state (rebit). Group-theoretically, many standard gate matrices have determinant -1 and therefore lie outside SO(N) when the real vector space has the same dimension as the complex one; continuous evolution from the identity can only remain inside the connected component SO(N). The standard 2N-dimensional real embedding is shown to be merely the matrix representation of complex endomorphisms, and the same restriction holds for the endomorphism algebra of real spaces of dimension 2^n suitable for n qubits. The conclusion is that complex numbers are necessary for the time evolution of quantum gates.

Significance. If the modeling assumptions hold, the manuscript supplies a dynamical and group-theoretic argument that complex structure is required by the continuous evolution of standard gates rather than being a mathematical convenience. It explicitly links the acquisition of complex phases to entanglement generation and identifies an obstruction in the real orthogonal group that cannot be circumvented by the usual real-to-complex isomorphism. The approach is parameter-free and relies only on standard properties of unitary evolution and Lie groups.

major comments (2)

[Section introducing the effective-Hamiltonian scheme] The modeling of every gate as U = exp(-i H τ) for a single fixed Hermitian H (the 'simple scheme' that produces latitude trajectories) is load-bearing for the necessity claim yet is introduced without demonstration that it captures the dynamics of gates realized by time-dependent controls, composite pulses, or adiabatic passages. Because the net unitary need not lie on a one-parameter subgroup, the argument that a real state vector must immediately leave its latitude line does not automatically extend to general continuous paths in U(N).
[Group-theoretic argument following the Bloch-sphere discussion] The group-theoretic obstruction rests on the observation that 'the matrices representing many quantum gates of interest have determinant -1.' Quantum gates are defined only up to global phase, so any det = -1 representative can be multiplied by a phase factor e^{iθ} to obtain a det = 1 element that lies in the connected component of U(N). The manuscript does not show why the physical equivalence class cannot always be represented by a det = 1 matrix reachable by continuous real orthogonal evolution, nor does it address the corresponding freedom in the real orthogonal group O(N).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable feedback on our manuscript. The comments highlight important aspects of our modeling assumptions and the handling of global phases. We address each major comment in detail below and indicate where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [Section introducing the effective-Hamiltonian scheme] The modeling of every gate as U = exp(-i H τ) for a single fixed Hermitian H (the 'simple scheme' that produces latitude trajectories) is load-bearing for the necessity claim yet is introduced without demonstration that it captures the dynamics of gates realized by time-dependent controls, composite pulses, or adiabatic passages. Because the net unitary need not lie on a one-parameter subgroup, the argument that a real state vector must immediately leave its latitude line does not automatically extend to general continuous paths in U(N).

Authors: We acknowledge that the simple scheme serves primarily as an illustrative example to demonstrate the acquisition of complex phases via latitude trajectories on the Bloch sphere. It is not claimed to encompass all possible implementations involving time-dependent Hamiltonians or composite pulses. The specific argument about leaving the latitude line is indeed tied to this one-parameter subgroup evolution. However, the broader necessity of complex numbers arises from the group-theoretic consideration that any continuous evolution starting from the identity in a real vector space must remain within the special orthogonal group SO(N), the connected component of O(N). We will revise the manuscript to better separate the illustrative simple scheme from the general group-theoretic obstruction, making clear that the latter applies to arbitrary continuous paths in the real orthogonal group and thus extends the necessity claim beyond the simple scheme. revision: partial
Referee: [Group-theoretic argument following the Bloch-sphere discussion] The group-theoretic obstruction rests on the observation that 'the matrices representing many quantum gates of interest have determinant -1.' Quantum gates are defined only up to global phase, so any det = -1 representative can be multiplied by a phase factor e^{iθ} to obtain a det = 1 element that lies in the connected component of U(N). The manuscript does not show why the physical equivalence class cannot always be represented by a det = 1 matrix reachable by continuous real orthogonal evolution, nor does it address the corresponding freedom in the real orthogonal group O(N).

Authors: This is a pertinent observation, and we appreciate the referee pointing it out. While a global phase can indeed adjust the determinant to +1 within U(N), the resulting matrix is typically complex. For a real quantum mechanical description, the evolution operator must be a real orthogonal matrix. In the projective equivalence class of the gate, the real representatives (when they exist) generally belong to the determinant -1 component of O(N). Consider the Hadamard gate with its standard real representative H satisfying det(H) = -1. The real matrices in its equivalence class are ±H, both of which have determinant -1 when N is even. Consequently, no real special orthogonal matrix represents this gate. Continuous evolution from the identity can only reach elements of SO(N), not the other component of O(N). We will incorporate a discussion of global phase freedom and its limitations in the real setting into the revised manuscript to address this explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses explicit modeling assumptions and standard group/representation facts without reduction to self-referential inputs.

full rationale

The paper explicitly introduces a modeling scheme (effective Hamiltonian generating gate action, producing latitude trajectories on the Bloch sphere relative to an eigenvector) and then applies standard results: continuous real evolution requires SO(N) matrices, many gates have det=-1 so cannot lie in SO(N) for equal dimension, and the 2N real embedding is an isomorphism reproducing complex matrix representations. No parameters are fitted and called predictions, no self-citations are load-bearing for the core argument, no ansatz is smuggled, and no uniqueness theorem from prior author work is invoked. The steps are self-contained against external mathematical benchmarks (unitary groups, determinants, Endomorphism spaces) and do not reduce the conclusion to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that gate action is continuous unitary evolution generated by a time-independent effective Hamiltonian and on standard facts about matrix groups and representations; no free parameters or new entities are introduced.

axioms (2)

domain assumption Quantum gate action corresponds to continuous time evolution under an effective Hamiltonian
Invoked to derive Bloch-sphere latitude trajectories from the Schrödinger equation.
standard math Unitary matrices with determinant -1 cannot lie in the connected component of the special orthogonal group in the same dimension
Standard fact from Lie group theory used to rule out real SO(n) representations.

pith-pipeline@v0.9.0 · 5655 in / 1504 out tokens · 54931 ms · 2026-05-10T06:26:10.033969+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 2 canonical work pages

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Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Thinh P Le, Armin Tavakoli, Nicolas Gisin, Antonio Ac´ ın, and Miguel Navascu´ es. Quantum theory based on real numbers can be experimentally falsified.Nature, 600(7890):625– 629, 2021

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Carlton M Caves, Christopher A Fuchs, and Pranaw Rungta. Entanglement of formation of an arbitrary state of two rebits.Foundations of Physics Letters, 14(3):199–212, 2001

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work page Pith review arXiv 2002

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