Quantum time dynamics employing the Yang-Baxter equation for circuit compression

Quantum time dynamics (QTD) is considered a promising problem for quantum supremacy on near-term quantum computers. However, QTD quantum circuits grow with increasing time simulations. This study focuses on simulating the time dynamics of one-dimensional (1D) integrable spin chains with nearest-neighbor interactions. We have proved the existence of a reflection symmetry in the quantum circuit employed for simulating the time evolution of certain classes of 1D Heisenberg model Hamiltonians by virtue of the quantum Yang-Baxter equation, and how this symmetry can be exploited to compress and produce a shallow quantum circuit. With this compression scheme, the depth of the quantum circuit becomes independent of step size and only depends on the number of spins. We show that the depth of the compressed circuit is rigorously a linear function of the system size for the studied Heisenberg model Hamiltonians in the present work. As a consequence, the number of CNOT gates in the compressed circuit only scales quadratically with the system size, which allows for the simulations of time dynamics of very large 1D spin chains. We derive the compressed circuit representations for different special cases of the Heisenberg Hamiltonian. We compare and demonstrate the effectiveness of this approach by performing simulations on quantum computers.

Figure 1

1D spin chain showing two families (orange and blue) in which all elements in each family commute within that family.

Figure 2

Quantum circuit for time evolution of $N$ spins, composed of $n$ alternative layers using the Trotter approximation.

Figure 3

(a) Quantum circuit representation of the YBE for three qubits. (b) Reflection symmetry is achieved by using the YBE four times on four qubits (the action of YBE on which triplets are shown by black dots).

Figure 4

Compression scheme for four qubits. Reflection symmetry exists with two layers of alternative gates. The addition of a third layer can be absorbed into the two layers by recursive usage of reflection symmetry (red bracket) via the YBE and merge identity (black dotted box).

Figure 5

Comparison of time dynamics for three spins with the $XY$ ($J_x=-0.8$ and $J_y=-0.2$) model Hamiltonian on the IBM (Manila) device (8192 shots) with and without compression. Results from the Qiskit simulator serve as a baseline. Note that for simulating the $XY$ model, each two-qubit gate includes two CNOTs, therefore the compressed circuit for the simulation only includes six CNOTs constituting three layers in comparison with 400 CNOTs without compression.

Figure 6

Reflection symmetry of the quantum circuit employed for simulating the time evolution of the Heisenberg model Hamiltonian defined in Table 1. (a) When the number of qubits $N$ is even, there exists a vertical symmetry for the first $N$ layers of the circuit. (b) When $N$ is odd, there exists a horizontal symmetry for the first $N$ layers of the circuit.

Figure 7

A diagrammatic proof of the reflection symmetry in the quantum circuit employed for simulating the time evolution of the 1D Heisenberg model with the Hamiltonian defined in Table 1. Here, the blue blocks denote the original two-qubit gates, the green blocks denote the two-qubit gates after YBE transformation, and the red dashed blocks denote the destinations that are connected by dashed arrow curves to the highlighted blue blocks after performing consecutive three-spin YBE transformations. The orange arrow indicates the direction for moving the two-qubit gates in each step.

Figure 8

In the time evolution circuit of a 10-qubit Heisenberg model, the top left two-qubit (blue block highlighted by a red frame) gate can be gradually moved downward to the bottom right position (denoted by the dashed red block) through eight consecutive three-site YBE operations (labeled by circled numbers). As a result of the eight consecutive three-site YBE operations, all the two-qubit gates in the diagonal direction (included in the red dashed frame on the right-hand side) need to update their rotations and phase factors.

Figure 9

Circuit compression for the $XX+Z$ model. Two-qubit gates for $XX$ interaction are denoted by blue blocks, and single-qubit rz gates are denoted by green blocks. Note that blue and green blocks commute.