The geometry of quantum computation

TL;DR

The paper reframes quantum circuit complexity as a problem in Riemannian geometry on SU(2^n), introducing a right-invariant metric that penalizes high-weight interactions and linking geodesic distance to gate complexity. It develops the full geometric toolkit—Levi-Civita connection, geodesic (Lax-type) equations, constants of motion, and analytic solutions—and introduces the lifted Jacobi equation and geodesic derivative to study geodesic deformation with changing penalties and endpoints. The authors demonstrate both exact and approximate geodesic solutions in special cases (e.g., constant-Hamiltonian geodesics and three-qubit systems), and present a practical deformation-based method to numerically find geodesics to a target unitary, along with a robust discussion of conjugate points and when geodesics cease to minimize globally. They also address fundamental barriers via Razborov-Rudich-type results and propose an ancilla-extension framework to extend geometric analysis to more general quantum circuits. Overall, the work lays a foundational link between geometry and quantum computation, offering methods and caveats for leveraging geometric insights while acknowledging limits imposed by complexity theory.

Abstract

Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.

Figures (3)

• Figure 1: Log plot of the absolute value of the minimum eigenvalue of $E_2$ ($\lambda_{\min}(E_2)$) versus time, for the transverse Ising model with external field $h=1$. The penalty parameter is in the regime of computational interest, $q=4^n=64$. Sharp dips indicate conjugate points.
• Figure 2: Geodesic deformation to a randomly-chosen unitary on $n=3$ qubits. Panel $(a)$ shows how the Pauli components of the initial dual Hamiltonian, $l_q^\sigma(0) = \hbox{tr}(\sigma L_q^\sigma(0))/2^n$, vary with the penalty parameter up to $q=4^n=64$. Of the $4^n=64$ Pauli components, only $16$ representatives are shown, for clarity, but all converge in the large $q$ limit. Blue lines are components where $\sigma \in \mathcal{P}$, red lines are where $\sigma \in \mathcal{Q}$. The inset shows how the length of the geodesic segment from $I$ to $U$ varies with $q$. Panel $(b)$ shows the minimum eigenvalue of the vectorized form of the propagator $\mathcal{J}_t$ as a function of time along the geodesic found for $q=64$. No conjugate points are evident. The inset shows the operator norm of the difference between the target unitary $U$, and $U(t)$ along the $q=64$ geodesic, showing that the target is indeed reached at the final time $T=1$
• Figure 3: Geodesic deformation to the quantum Fourier transform on $n=3$ qubits. Panel $(a)$ shows how the Pauli components of the initial dual Hamiltonian, $l_q^\sigma(0) = \hbox{tr}(\sigma L_q^\sigma(0))/2^n$, vary with the penalty parameter up to $q=16$. The inset shows how the length of the geodesic segment from $I$ to $U$ varies with $q$. Blue lines are components where $\sigma \in \mathcal{P}$, red lines are where $\sigma \in \mathcal{Q}$. Panel $(b)$ shows the minimum eigenvalue of the propagator $\mathcal{J}_t$ as a function of time along the final geodesic found for $q=16$. The sharp dip at the final time $T=1$ indicates a conjugate point. The inset shows the operator norm of the difference between the target unitary $U$, and $U(t)$ along the $q=16$ geodesic, showing that the target is indeed reached at the final time $T=1$

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Proposition 1