RedCarD: A Quantum Assisted Algorithm for Fixed-Depth Unitary Synthesis via Cartan Decomposition

TL;DR

The paper addresses synthesizing fixed-depth quantum circuits for unitaries of the form $U(t)=e^{-itH}$ by leveraging Cartan decompositions of the dynamical Lie algebra $\mathfrak{g}(H)\subseteq \mathfrak{su}(2^n)$. It introduces Reductive Cartan Decomposition (RedCarD), which partitions the $\mathfrak{k}$-part into a sequence of subspaces $\mathfrak{k}^{r}_{1\dots r-1}(\mathfrak{b})$ relative to an Abelian subalgebra $\mathfrak{b}$ and optimizes over progressively fewer parameters by solving independent subproblems. The authors develop a reductive KHK framework with theorems ensuring that the transformed Hamiltonian enters progressively smaller non-Abelian subspaces, enabling a fixed-depth circuit $U(t)=K^1_c \cdots K^{|\,\tilde{\mathfrak{h}}|}_c e^{-i h t} K^{|\,\tilde{\mathfrak{h}}|\dagger}_c \cdots K^{1\dagger}_c$, and demonstrate a quantum-assisted cost evaluation using a TFIM on IBM and Quantinuum hardware. The work shows substantial reductions in classical optimization overhead and validates the approach with hardware experiments, highlighting both the potential and the remaining scalability challenges due to exponential growth of the dynamical Lie algebra for generic models.

Abstract

A critical step in developing circuits for quantum simulation is to synthesize a desired unitary operator using the circuit building blocks. Studying unitaries and their generators from the Lie algebraic perspective has given rise to several algorithms for synthesis based on a Cartan decomposition of the dynamical Lie algebra. For unitaries of the form $e^{-itH}$, such as time-independent Hamiltonian simulation, the resulting circuits have depth that does not depend on simulation time $t$. However, finding such circuits has a large classical overhead in the cost function evaluation and the high dimensional optimization problem. In this work, by further partitioning the dynamical Lie algebra, we break down the optimization problem into smaller independent subproblems. Moreover, the resulting algebraic structure allows us to easily shift the evaluation of the cost function to the quantum computer, further cutting the classical overhead of the algorithm. As an application of the new hybrid algorithm, we synthesize the time evolution unitary for the 4-site transverse field Ising model on several IBM devices and Quantinuum's H1-1 quantum computer.

Figures (13)

• Figure 1: (a) Schematic for the Cartan decomposition of the DLA and subsequent fragmentation of $\mathfrak{k}$. Here $\mathfrak{k}^r_{1\dots r-1}$ is the subspace spanned by the Pauli strings that commute with the first $r-1$ basis elements of $\mathfrak{h}$ and anticommute with the $r^{\text{th}}$ basis element of $\mathfrak{h}$ (see \ref{['def:ksets']}). (b) Fixed-depth circuit compilation of the unitary $e^{-itH}$. Each unitary block $K^r_c$ is a product of the exponentials of Pauli strings in $\mathfrak{k}^r_{1\dots r-1}$ (see \ref{['eq:prodkr']}).
• Figure 2: Number of parameters to optimize over for each step of the reductive KHK decomposition for the 1-dimensional nearest neighbor XY, transverse field Ising/XY, and Heisenberg spin models.
• Figure 3: Runtime and cost function calls comparison between the standard algorithm and the modified version. Statistics are done on 10 converging runs. A run converges if the ratio of the Hilbert-Schmidt norm of the part of $K^\dagger H K$ that lies outside of $\mathfrak{h}$ to the Hilbert-Schmidt norm of $H$ reaches 1% in less than $10^5$ iterations. The same ansatz is used for each run, and initial angles are randomized. We used the Rotosolve algorithm ostaszewski2021structure to minimize the cost function.
• Figure 4: Circuit for a 4-site TFIM time evolution unitary. An arrow doublet connecting two qubits $p$ and $q$ depicts the unitary $e^{i\alpha \widehat{X_pY_q}+i\beta \widehat{Y_pX_q}}$. Flipping the doublet represents Hermitian conjugation. The color of a dashed box indicates the basis element of $\mathfrak{h}$ with which the doublets do not commute.
• Figure 5: Quantum assisted KHK decomposition. At each step, only the angles in $K^r$ are varied as the previous angles are already fixed from the previous optimizations.

Theorems & Definitions (25)

• Theorem 1: Improved KHK Decomposition
• Definition 1
• Lemma 1: Reductive Cartan Decomposition
• Definition 2
• Theorem 2: Reductive KHK Decomposition
• Definition 3
• Theorem 3
• Definition B.1
• Theorem B.1
• Corollary B.1