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arXiv:2511.16736

Unitary synthesis with optimal brick wall circuits

David Wierichs, Korbinian Kottmann, Nathan Killoran

quant-phSubmitted Nov 20, 2025• Announced Nov 24, 2025• 0 citations
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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.

Comments
15 pages, 12 figures, 4 tables. Associated code: https://github.com/XanaduAI/unicirc/ . Comments welcome!

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We present quantum circuits with a brick wall structure using the optimal number of parameters and two-qubit gates to parametrize SU(2n)SU(2^n)SU(2n), and provide evidence that these circuits are universal for n≤5n\leq 5n≤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(2n)SU(2^n)SU(2n) for n=3,4,5n=3, 4, 5n=3,4,5, and we extend it to the subgroups SO(2n)SO(2^n)SO(2n) and Sp∗(2n)Sp^\ast(2^n)Sp∗(2n). 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.
We introduce the flag decomposition as a central tool for unitary synthesis. It lets us carve out a diagonal unitary with 2n2^n2n degrees of freedom in such a way that the remaining flag circuit is parametrized by the optimal number of 4n−2n4^n-2^n4n−2n rotations. This enables us to produce parameter-optimal quantum circuits for generic unitaries and matrix product state preparation. Our approach improves upon the state of the art, both when compiling down to the {Clifford + Rot} gate set with what we call selective de-multiplexing, and when using phase gradient resource states together with quantum arithmetic to implement multiplexed rotations. All of our synthesis methods are efficiently implementable in terms of recursive Cartan decompositions realized by standard linear algebra routines, making them applicable to all practically relevant system sizes.
Although with some yet severe limitations, Physical working Quantum Computers are becoming available for end users. Such devices are based on the rules of Quantum Mechanics, which state that physical systems evolve through unitary transformations. However, as in the classical case, in order to have a model of computation, such unitary evolutions are expressed/approximated inside Quantum Computers in terms of a finite set of one/two qubit operations (i.e., through a universal set of gates). Single qubit gates are fault-tolerant, while the same cannot be said for two qubit gates. Hence, unitary matrices adopted in Quantum Algorithms must be synthesized in terms of this universal set of operations to obtain a quantum circuit. In such synthesis procedure we prefer circuits with minimum number of qubits and with minimum circuit depth. Clifford+T universal set is one of the most adopted in the literature for synthesis. In such set we have 3 single qubit gates and the CNOT, which is a two qubit gate. Many efforts have been directed to devise algorithms that synthesize general unitary matrices into Clifford+T circuits. These algorithms usually tend to optimize circuit depth or eventually the number of T gates. Since two qubit gates are not fault tolerant, in this work we propose an ASP based technique to minimize the number of CNOT gates inside a Clifford+T circuit. Moreover, in real world quantum computers, qubit are usually connected to each other according to some particular topology, thus providing further constraints. Two qubit gates —hence, CNOT gate— can be directly applied only to pair of gates that are connected. Such constraint has to be taken into account during the synthesis of CNOT minimal circuits. We propose an ASP model to solve the problem of synthesizing CNOT minimal circuits under topological constraints. We provide experimental evidence of the scalability of our proposal.
We study brick wall quantum circuits enjoying a global fermionic symmetry. The constituent 2-qubit gate, and its fermionic symmetry, derive from a 2-particle scattering matrix in integrable, supersymmetric quantum field theory in 1+1 dimensions. Our 2-qubit gate, as a function of three free parameters, is of so-called free fermionic or matchgate form, allowing us to derive the spectral structure of both the brick wall unitary U_F U F and its, non-trivial, Hamiltonian limit H_{\gamma} H γ in closed form. We find that the fermionic symmetry pins H_{\gamma} H γ to a surface of critical points, whereas breaking that symmetry leads to non-trivial topological phases. We briefly explore quench dynamics for this class of circuits.
Matchgate unitaries are ubiquitous in quantum computation due to their relation to non-interacting fermions and because they can be used to benchmark quantum computers. Implementing such unitaries on fault-tolerant devices requires first compiling them into a discrete universal gate set, typically Clifford+T+T+T. Here, we propose a different approach for their synthesis: compile matchgate unitaries using only matchgate gates. To this end, we first show that the matchgate-Clifford group (the intersection of the matchgate and Clifford groups) plus the T‾\overline{T}T gate (a TTT unitary up to a phase) is universal for the matchgate group. Our approach leverages the connection between nnn-qubit matchgate circuits and the standard representation of SO(2n)\mathbb{SO}(2n)SO(2n), which reduces the compilation from 2n×2n2^n\times 2^n2n×2n unitaries to 2n×2n2n\times2n2n×2n ones, thus reducing exponentially the size of the target matrix. Moreover, we rigorously show that this scheme is efficient, as an approximation error εSO(2n)\varepsilon_{\mathbb{SO}(2n)}εSO(2n)​ incurred in this smaller-dimensional representation translates at most into an O(n εSO(2n))O(n \,\varepsilon_{\mathbb{SO}(2n)})O(nεSO(2n)​) error in the exponentially large unitary. In addition, we study the exact version of the matchgate synthesis problem, and we prove that all matchgate unitaries UUU such that U⊗U∗U\otimes U^*U⊗U∗ has entries in the ring Z[1/2,i]\mathbb{Z}\big[1/\sqrt 2,i\big]Z[1/2​,i] can be exactly synthesized by a finite sequence of gates from the matchgate-Clifford+T‾+\overline{T}+T set, without ancillas. We then use this insight to map optimal exact matchgate synthesis to Boolean satisfiability, and compile the circuits that diagonalize the free-fermionic XXXXXX Hamiltonian on n=4, 8n=4,\,8n=4,8 qubits.
We present new optimal and heuristic algorithms for exact synthesis of multi-qubit unitaries and isometries. For example, our algorithms find Clifford and T circuits for unitaries with entries in Z[i,1/2]\mathbb{Z}[i,1/\sqrt{2}]Z[i,1/2​]. The optimal algorithms are the A* search instantiated with a new data structure for graph vertices and new consistent heuristic functions. We also prove that for some gate sets, best-first search synthesis relying on the same heuristic is efficient. For example, for two-qubit Clifford and T circuits, our best-first search runtime is proportional to the T-count of the unitary. Our algorithms rely on Hermite and Smith Normal Forms of matrices with entries in a ring of integers of a number field, and we leverage the theory of and algorithms for Hermitian lattices over number fields to prove efficiency. These new techniques are of independent interest for future work on multi-qubit exact circuit synthesis and related questions.
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