Faster Digital Quantum Simulation by Symmetry Protection

Simulating the dynamics of quantum systems is an important application of quantum computers and has seen a variety of implementations on current hardware. We show that by introducing quantum gates implementing unitary transformations generated by the symmetries of the system, one can induce destructive interference between the errors from different steps of the simulation, effectively giving faster quantum simulation by symmetry protection. We derive rigorous bounds on the error of a symmetry-protected simulation algorithm and identify conditions for optimal symmetry protection. In particular, when the symmetry transformations are chosen as powers of a unitary, the error of the algorithm is approximately projected to the so-called quantum Zeno subspaces. We prove a bound on this approximation error, exponentially improving a recent result of Burgarth, Facchi, Gramegna, and Pascazio. We apply the symmetry-protection technique to the simulations of the $XXZ$ Heisenberg interactions with local disorder and the Schwinger model in quantum field theory. For both systems, the technique can reduce the simulation error by several orders of magnitude over the unprotected simulation. Finally, we provide numerical evidence suggesting that the technique can also protect simulation against other types of coherent, temporally correlated errors, such as the $1/f$ noise commonly found in solid-state experiments.

Popular Summary

Simulating quantum systems is an important application of quantum computers. However, while the dynamics of a target quantum system is continuous, its approximation on a quantum computer is often made by breaking up the simulation into small, finite time steps. Then, in each time step, a discrete series of elementary quantum gates realize an approximate version of the evolution. This discreteness introduces undesirable errors to the simulation. In contrast to dephasing and other sources of incoherent noise, this type of “coherent” error does not represent any measurement or other information flow from the simulator to the environment, which can often be removed by a variety of correction techniques. Here, we propose a technique that uses the symmetries of the target system to suppress this simulation error and thus significantly reduce the gate count of the simulation.

Specifically, we insert rotations generated by the symmetries of the target system in between the discrete time steps of the simulation. While these rotations do not affect the true evolution, they induce destructive interference between the coherent errors from different steps of the simulation, resulting in a smaller total error, fewer quantum gates, and hence a faster quantum simulation. We make a connection to the quantum Zeno effect and prove bounds that exponentially improve the state of the art. By applying our technique to the simulations of several quantum systems, including spin chains and lattice quantum field theories, we demonstrate significant error reduction due to the symmetry protection, sometimes by several orders of magnitude.

Figure 1

For algorithms that simulate the dynamics of quantum systems by decomposing the evolutions into many time steps, we interweave the corresponding simulation circuits (blue) with unitary transformations generated by the symmetries of the systems (orange). These transformations protect the simulations against errors that violate the symmetries, resulting in faster and more accurate simulations.

Figure 2

The total error of the simulation is analogous to the average distance a walker walks in $r$ steps of the simulation. In each time step, the walker walks a small distance along a vector representing the error operator in the space of operators. (a) Without any symmetry protection, the walker keeps walking towards almost the same direction, resulting in a total distance that scales linearly with the number of steps $r$, corresponding to the total error scaling as $O\left(1\right)$. (b) The symmetry transformations make the walker walk in a possibly different direction in every time step. When the direction is uniformly random (see Sec. 4a 4a1 and Fig. 3 for an example), the total distance only scales as $O\left(\sqrt{r}\right)$, resulting in the total error scaling as $O\left(1/\sqrt{r}\right)$. (c) Sometimes, it is possible to design an optimal set of symmetry transformations that makes the walker return to the origin [see Eq. (38) for an example], resulting in an $O\left(1/r\right)$ error for the entire simulation.

Figure 3

The total error in simulating the Hamiltonian Eq. (34) at $n=4$ for a fixed evolution time $t=1$ as a function of the Trotter number $r$ using four different schemes: the raw first-order Trotterization (“Raw”), the first-order Trotterization protected by a random set symmetry transformation (“SP-Rand”), the first-order Trotterization protected by the optimal set in Eq. (38) (“SP-Det”), and the random-ordering scheme in Ref. [12] (“Random Ordering”). We indicate the scalings obtained from power-law fits to the right of the plot. We repeat the simulation 100 times, each with a different set of randomly generated interactions $J_{ij}$. The dots correspond to the median of the errors at each value of $r$ and the bars represent the corresponding 25%–75% percentiles regions.

Figure 4

The number of Trotter steps required for the simulation of $n$ qubits evolved under Eq. (40) for time $t=n$ to meet a fixed-error tolerance $\epsilon =0.01$. We compare this Trotter number of a simulation without any symmetry protection (“Raw”, blue) and a simulation with random symmetry protection (“SP”, orange) at $h=2$ (left panel) and $h=8$ (right panel), which correspond to the system being in the ETH and the MBL, respectively. The dashed lines are the linear fits of the data in the log-log scale. The simulation is repeated 100 times with different instance of the disorder $h_i$. The dots represent the median of the Trotter number at each $n$ and the error bars correspond to the 25%–75% percentile region. The numerics show that symmetry protecting the simulation reduces the number of Trotter steps, and hence the gate count, by about 2 to 4 times in both the ETH and the MBL phases.

Figure 5

The required number of Trotter steps in simulating the Hamiltonian Eq. (40) of $n=8$ qubits for time $t=n$ as a function of the disorder strength in an unprotected simulation (“Raw”, blue) and in a symmetry-protected simulation (“SP”, orange). Each dot represents the median Trotter number over 100 different instances of the random fields. The bars correspond to the 25%–75% percentile region.

Figure 6

The probability for the final state to leak outside the physical subspace due to Trotter errors in simulating the Schwinger model. We consider simulations without symmetry protection (blue) and with symmetry protection under different schemes: uniform sets of transformations drawn from ${\mathbb{Z}}_8$ (red) and U(1) (orange) and random sets of transformations drawn from ${\mathbb{Z}}_8$ (purple) and U(1) (green). The purple and green areas overlap each other almost completely. The dots correspond to the median and the shaded areas correspond to the 25%–75% percentile of 100 repetitions.

Figure 7

The leakage to the unphysical subspace as a function of time in simulating the Schwinger model using advanced algorithms. We consider a raw simulation (blue), a simulation protected by a random set of transformations drawn from the U(1) symmetry group (green), and a simulation protected by a uniform set of transformations (orange). The solid dots correspond to the median of 100 repetitions and the shaded area corresponds to the 25%–75% percentile.

Figure 8

The leakage probability due to experimental noise as a function of time at different values of the correlation length $\lambda$. The simulation is repeated 100 times with different instances of the experimental noise. The solid dots represent the median of the leakage and the bars correspond to the 25%–75% percentile regions.

Figure 9

The frequent kicks confine the dynamics of the system (solid arrows) to the so-called quantum Zeno subspaces, defined by the projectors $P_{\mu }$ in the spectral decomposition of the kicks $U_{\mathrm{kick}}=\sum _{\mu }{e}^{-i{\varphi }_{\mu }}{P}_{\mu }$. In particular, the kicks suppress the probability for the system to travel between the subspaces (dashed arrow). By generating the kicks from the symmetries of the system, we can target the simulation error—the sole contributor to possible violations of the symmetries in an ideal simulation—for suppression.