SU(2) non-Abelian gauge field theory in one dimension on digital quantum computers

A dynamical quantum simulation of SU(2) non-Abelian gauge field theory on a digital quantum computer is presented. This was enabled on current quantum hardware by introducing a mapping of the field onto a register of qubits that utilizes local gauge symmetry while preserving local constraints on the fields, reducing the dimensionality of the calculation. Controlled plaquette operators and gauge-variant completions in the unphysical part of the Hilbert space were designed and used to implement time evolution. The new techniques developed in this work generalize to quantum simulations of higher dimensional gauge field theories.

Figure 1

(top) Diagram of the lattice distribution of $\lceil {\log }_2(2{\mathrm{\Lambda }}_j+1)\rceil$-qubit registers and the action of $\widehat{\square }$ on SU(2) plaquettes in one dimension. $\widehat{\square }$ operates on the four qubit registers in the plaquette and is controlled by the four neighboring qubit registers to enforce the Gauss’s law constraint. (bottom) The plaquette operator with labeled angular momentum registers.

Figure 2

Digital circuit implementation of the plaquette operator centered on $j_a$ for a truncated lattice with ${\mathrm{\Lambda }}_j=1/2$, two plaquettes, and PBCs as depicted at the right. The angles $\tilde{\beta }$ defining this circuit are given in Eq. (b1) to be $\tilde{\beta }=(3/8,5/8)$.

Figure 3

(top) Expectation value of the electric energy contribution of the first plaquette in the two-plaquette lattice with PBCs and coupling $g^2=0.2$ computed on IBM’s Tokyo. The dashed(purple) and dot-dashed(blue) lines are the $N_{\text{Trot}}=1$, 2 Trotterized expectation values, while the thick gray line is the exact time evolution. (bottom) Measured survival probability to remain in the physical subspace for one and two trotter steps, $N_{\text{Trot}}$, and one and two $r$ values indicating stochastically inserted $2r-1$ CNOTs per CNOT in the digital implementation. Uncertainties represent statistical variation, as well as a systematic uncertainty estimated from reproducibility measurements. The icons (defined in Ref. [48]) denote computations performed on quantum devices.

Figure 4

Digital circuit implementation of the plaquette operator centered on $j_a$ for a truncated lattice with ${\mathrm{\Lambda }}_j=1/2$. The circuit elements appearing in this circuit are the Hadamard, CNOT, and Z-axis single-qubit rotation implementing a Z-to-X basis change, a controlled bit flip, and a relative phase, respectively.