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Unitary synthesis with optimal brick wall circuits

David Wierichs, Korbinian Kottmann, Nathan Killoran

TL;DR

The paper develops brick-wall quantum circuit templates that achieve the minimal resource costs for universal parameterization of $SU(2^n)$, with extensions to $SO(2^n)$ and $Sp^\ast(2^n)$, and provides numerical evidence of universality for $n=3,4,5$ via full-rank Jacobian tests and expressibility analyses. The approach combines a constructive ansatz characterization with space-reduction and a Jacobian-rank necessary condition to identify universal candidates, complemented by practical unitary synthesis, expressibility assessments, and applications to vibronic and fast-forwardable Hamiltonians. While a full analytic universality proof is contingent on a conjecture about the absence of walls in the parameter space, the results show state-of-the-art circuit efficiency (optimal parameter and CZ counts) and robust numerical support for the proposed templates. The work has practical implications for efficient unitary synthesis and fault-tolerant implementations, offering a scalable route to high-dimensional quantum dynamics using compact, highly expressive parameterizations.

Abstract

We present quantum circuits with a brick wall structure using the optimal number of parameters and two-qubit gates to parametrize $SU(2^n)$, and provide evidence that these circuits are universal for $n\leq 5$. For this, we successfully compile random matrices to the presented circuits and show that their Jacobian has full rank almost everywhere in the domain. Our method provides a new state of the art for synthesizing typical unitary matrices from $SU(2^n)$ for $n=3, 4, 5$, and we extend it to the subgroups $SO(2^n)$ and $Sp^\ast(2^n)$. We complement this numerical method by a partial proof, which hinges on an open conjecture that relates universality of an ansatz to it having full Jacobian rank almost everywhere.

Unitary synthesis with optimal brick wall circuits

TL;DR

The paper develops brick-wall quantum circuit templates that achieve the minimal resource costs for universal parameterization of , with extensions to and , and provides numerical evidence of universality for via full-rank Jacobian tests and expressibility analyses. The approach combines a constructive ansatz characterization with space-reduction and a Jacobian-rank necessary condition to identify universal candidates, complemented by practical unitary synthesis, expressibility assessments, and applications to vibronic and fast-forwardable Hamiltonians. While a full analytic universality proof is contingent on a conjecture about the absence of walls in the parameter space, the results show state-of-the-art circuit efficiency (optimal parameter and CZ counts) and robust numerical support for the proposed templates. The work has practical implications for efficient unitary synthesis and fault-tolerant implementations, offering a scalable route to high-dimensional quantum dynamics using compact, highly expressive parameterizations.

Abstract

We present quantum circuits with a brick wall structure using the optimal number of parameters and two-qubit gates to parametrize , and provide evidence that these circuits are universal for . For this, we successfully compile random matrices to the presented circuits and show that their Jacobian has full rank almost everywhere in the domain. Our method provides a new state of the art for synthesizing typical unitary matrices from for , and we extend it to the subgroups and . We complement this numerical method by a partial proof, which hinges on an open conjecture that relates universality of an ansatz to it having full Jacobian rank almost everywhere.

Paper Structure

This paper contains 24 sections, 4 theorems, 27 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Derivation
  4. Ansatz characterization
  5. Reducing the search space
  6. Necessary condition: Jacobian rank
  7. Selecting universal candidates
  8. Minimizing the parameter count
  9. Numerical results
  10. Expressibility by Sim et al.
  11. Unitary synthesis of typical matrices
  12. Applications
  13. Vibronic Hamiltonians
  14. Fast-forwardable Hamiltonians
  15. Mathematical considerations
  16. ...and 9 more sections

Key Result

Lemma 0

If the image of a PQC is dense in $\mathcal{G}$, the image is the full group.

Figures (12)

  • Figure 1: Brick wall circuit structure with the optimal number of parameters and two-qubit gates, which we claim to yield a universal circuit. The number of times, $n_\ell$, that the green box is repeated is given in \ref{['eq:nell']}. The "remainder" block is constructed such that the minimal number of parameters is used. \ref{['tab:SUN']} shows the explicit structure for $n\in \{3, 4, 5\}$.
  • Figure 2: Numerical expressibility test from Sim-Johnson-Aspuru-Guzik for the parameter-optimal circuit on three qubits. We randomly sample $K$ pairs of parameter vectors, compute the fidelities between the states prepared by the ansatz, and compare the resulting fidelity distribution to that of Haar random unitaries. The expressibility $\operatorname{Expr}$ of the ansatz is defined as the Kullback-Leibler divergence between them, with smaller values indicating larger expressibility. a-c) Fidelity distributions for the circuit (histogram) and for Haar random unitaries (solid line), for $K=10^4,10^5,10^6$, respectively. d) Expressibility $\operatorname{Expr}$ for $10^2\leq K\leq 10^6$ (markers). The numerical value depends on $K$, and we observe a steady convergence towards $0$ with scaling $\mathcal{O}(K^{-0.9})$ (solid line), which we interpret as an indication for universality. See \ref{['fig:expressibility_app']} for the cases $n=4, 5$.
  • Figure 3: Compiling $10$ Haar random unitaries by starting variational optimization from $10$ random initial values each, sampled from a normal distribution with $\mu=0, \sigma=\tfrac{1}{5}$. Each target unitary converges at least once within the $10$ trials to the threshold of $10^{-10}$ (dotted line). $\kappa$ in the lower left corner indicates the overall success rate over the $10\times10$ trials. The used target precision is $10^{-12}$ (dashed line), allowing for the more detailed analysis of $\kappa$ in \ref{['fig:successprob']}.
  • Figure 4: Success rates $\kappa$ from the $100$ compilation attempts in \ref{['fig:convergence']} for different threshold values $\epsilon$. Curiously, for threshold values above $10^{-10}$, the success probability increases with the qubit number. One potential explanation is that the local minima encountered for higher qubit counts provide approximations with higher accuracy.
  • Figure 5: Total compilation CPU time for $1000$ ($500$, $100$) random target unitaries on $n=3$ ($n=4$, $n=5$) qubits. Variational optimizations are run with randomized initial parameters until the first run that achieves a cost below $\epsilon=10^{-10}$. We also report the average time to compilation $\mu$, but note the large variance between different targets.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Lemma 0
  • Lemma 0
  • Conjecture 1
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 1
  • proof