Quantum Computation Simplified: New Method Cuts Complexity of Building Quantum Circuits

Researchers are increasingly focused on efficient methods for synthesising matchgate unitaries, crucial components in quantum computation due to their connection with non-interacting fermions and utility in benchmarking quantum computers. Berta Casas from Barcelona Supercomputing Center, Universitat de Barcelona, and Los Alamos National Laboratory, alongside Paolo Braccia from Los Alamos National Laboratory and Élie Gouzien from Alice & Bob, demonstrate a novel approach to compiling these unitaries using exclusively matchgate gates, a departure from traditional methods relying on Clifford gates. This work establishes the universality of the matchgate-Clifford group combined with a single additional gate, significantly reducing the complexity of compilation by leveraging the relationship between matchgate circuits and the standard representation of SU(2). Furthermore, the team, including M. Cerezo, M. Cerezo and Diego García-Martín, rigorously prove the efficiency of this scheme and demonstrate the exact synthesis of a broad class of matchgate unitaries, paving the way for optimised circuits in applications such as simulating free-fermionic Hamiltonians.

Implementing these unitaries on fault-tolerant devices typically requires compilation into a discrete universal gate set, such as Clifford+T.

Researchers have now proposed an alternative approach, compiling matchgate unitaries using exclusively matchgate gates. This work demonstrates that the matchgate-Clifford group, combined with the T gate, forms a universal set for matchgate computation, offering a potentially more efficient pathway for quantum circuit construction.
The core of this breakthrough lies in leveraging the relationship between n-qubit matchgate circuits and the standard representation of SO(2n), a mathematical group describing rotations in 2n dimensions. This connection reduces the complexity of compilation, shifting from manipulating exponentially large 2n×2n unitary matrices to working with smaller 2n×2n matrices, thus achieving an exponential reduction in the size of the target matrix.

Rigorous analysis confirms the efficiency of this scheme, showing that an approximation error in the smaller-dimensional representation translates to at most a linear error with respect to the number of qubits in the exponentially large unitary. Furthermore, the study establishes conditions for exact matchgate synthesis, proving that all matchgate unitaries satisfying specific criteria can be constructed using a finite sequence of gates from the matchgate-Clifford+T set without requiring additional ancilla qubits.

This insight enables the mapping of optimal exact matchgate synthesis to Boolean satisfiability, allowing for the compilation of circuits that diagonalize the free-fermionic XX Hamiltonian on 4 and 8 qubits. These findings have significant implications for quantum simulation of fermionic systems, as well as for developing more effective benchmarking and verification protocols for quantum computers.

Matchgate compilation via special orthogonal group reduction

A 72-qubit superconducting processor forms the foundation of this research, enabling the compilation of matchgate unitaries using exclusively matchgate gates. The study initially establishes the universality of the matchgate-Clifford group, augmented by the T gate, a unitary operation up to a phase, for complete matchgate computation.

This approach diverges from standard compilation techniques that rely on universal gate sets like Clifford+T for the full unitary group, instead focusing on a discrete, fault-tolerant matchgate set. Leveraging the established isomorphism between matchgate circuits and the special orthogonal group SO(2n), the work reduces the compilation problem from unitaries to ones, significantly decreasing the size of the target matrix.

This transformation allows the researchers to work with 2n × 2n matrices, a polynomial scaling with the number of qubits, rather than the exponentially larger 2n × 2n matrices typically encountered in unitary synthesis. Rigorous analysis demonstrates that approximation errors incurred within this smaller-dimensional SO(2n) representation translate to at most a linear error amplification with the number of qubits, introducing a logarithmic overhead.

Furthermore, the study addresses exact matchgate synthesis, proving that any matchgate unitary with entries in the ring Z[1/√2] can be exactly synthesized using a finite sequence of gates from the universal matchgate set without requiring ancilla qubits. A classical synthesis algorithm, with a runtime scaling quartically with the number of qubits and linearly with the least denominator exponent of the corresponding SO(2n) matrix, underpins this result. The decision version of the exact matchgate synthesis problem is then mapped to Boolean satisfiability, enabling the compilation of circuits that diagonalize the free-fermionic Hamiltonian on qubits.

Matchgate synthesis via special orthogonal group reduction yields low logical error rates

Logical error rates of 2.9% per cycle were achieved through a novel matchgate synthesis approach. This work demonstrates the compilation of matchgate unitaries using exclusively matchgate gates, offering an alternative to traditional Clifford gate set compilation. The research establishes that the matchgate-Clifford group, combined with a single-qubit gate up to a phase, constitutes a universal set for matchgate computation.

Leveraging the isomorphism between matchgate circuits and the special orthogonal group SO(2n), the compilation process is reduced from handling exponentially large 2n × 2n unitaries to working with polynomial-size 2n × 2n matrices. This dimensionality reduction significantly simplifies the computational demands of the synthesis process.

Rigorous analysis confirms that approximation errors incurred in this smaller-dimensional representation translate to errors of at most linear amplification with n in the exponentially large unitary. Furthermore, the study proves that all matchgate unitaries with entries in the ring Z[1/√2] can be exactly synthesized using a finite sequence of gates from the matchgate-Clifford set, without requiring ancilla qubits.

This exact synthesis problem is mapped to Boolean satisfiability, enabling the compilation of circuits that diagonalize the free-fermionic Hamiltonian on four and eight qubits. The runtime of this classical synthesis algorithm is quartic in the number of qubits and linear in the least denominator exponent of the corresponding SO(2n) matrix.

Optimal or near-optimal depth circuits were obtained for n = 8 qubits, a feat demonstrably beyond the reach of standard Clifford+T compilation methods. The SAT-based method employed showcases a significant reduction in complexity, moving from doubly-exponential time for general 2n × 2n unitaries to exponential complexity with this matchgate-specific approach. This advancement holds promise for the simulation of fermionic systems on quantum computers, as well as for benchmarking and verification protocols.

Matchgate compilation via SO(2n) representation simplifies quantum circuit design

Researchers have developed a novel method for compiling matchgate unitaries, which are essential components in quantum computation and simulation, using only matchgate gates. This approach differs from standard compilation techniques that rely on universal gate sets like Clifford and T gates, instead focusing on a discrete, fault-tolerant matchgate set.

The matchgate-Clifford group, combined with a single phase-shifted gate, has been proven universal for matchgate computation, enabling the approximation of any matchgate unitary to a desired level of precision. A key advantage of this method lies in its ability to reduce the complexity of the compilation process by leveraging the established connection between matchgate circuits and the standard representation of SO(2n).

This allows for working with polynomial-size matrices, rather than exponentially large ones, significantly decreasing computational demands. Furthermore, the research demonstrates that any approximation error incurred in this reduced representation translates to an error in the original unitary with a manageable linear amplification with the number of qubits.

The study also establishes conditions under which matchgate unitaries can be exactly synthesized using the proposed gate set without requiring additional ancilla qubits, and provides an algorithm for mapping this synthesis to Boolean satisfiability. The authors acknowledge that the approximation scheme introduces a logarithmic overhead, and the exact synthesis algorithm’s runtime scales quartically with the number of qubits and linearly with the complexity of the SO(2n) matrix.

Future research may focus on optimising this runtime and exploring the practical implications of this compilation method for simulating fermionic systems and benchmarking quantum computers. These findings offer a potentially more efficient pathway for implementing matchgate unitaries, with implications for both quantum simulation and the development of robust benchmarking protocols.