Improving quantum simulation efficiency of final state radiation with dynamic quantum circuits

B. Nachman et al. [Phys. Rev. Lett. 126, 062001 (2021)] recently introduced an algorithm (QPS) for simulating parton showers with intermediate flavor states using polynomial resources on a digital quantum computer. We make use of a new quantum hardware capability called dynamical quantum computing to improve the scaling of this algorithm, which significantly improves the method precision. In particular, we modify the quantum parton shower circuit to incorporate midcircuit qubit measurements, resets, and quantum operations conditioned on classical information. This reduces the computational depth from $\mathcal{O}(N^5{\log }_2(N{)}^2)$ to $\mathcal{O}(N^3{\log }_2(N{)}^2)$ and the qubit requirements from $\mathcal{O}(N{\log }_2(N))$ to $\mathcal{O}(N)$. Using “matrix product state” state vector simulators, we demonstrate that the improved algorithm yields expected results for 2, 3, 4, and 5-steps of the algorithm. We compare absolute costs with the original QPS algorithm, and show that dynamical quantum computing can significantly reduce costs in the class of digital quantum algorithms representing quantum walks (which includes the QPS). Python code that implements QPS, both with and without dynamic gates, is publicly available on Github.

Figure 1

Dynamic computing workflow. The essential procedure consists of four steps: (1) Measure a subset of quantum resources in the QPU, represented here by the $k$-qubit register, (2) On the CPU perform some processing on the measured data, (3) Reset the measured qubits to the $|0⟩$ state so they can be reused, and (4) Based on CPU outputs, apply additional quantum operations on the QPU. Note that the QPU must maintain coherence throughout this procedure.

Figure 2

High-level circuit diagram of the first two steps of the QPS algorithm. Round gates indicate control qubits.

Figure 3

High-level circuit diagram of the first two steps of the improved QPS algorithm. Round gates indicate control qubits. Double wires indicate classical information stored on a CPU, and measured from $|h⟩$. Double wires attached to $U_p$ indicate that subgates in $U_p$ are selectively applied based on measured values of $h_0,h_1,\dots$. The $|0⟩$ attached to the same gate indicates a reset to the $|0⟩$ state.

Figure 4

Qubit cost comparison between the original QPS and the improved version with midcircuit measurements. This plot is simply an illustration of Eq. (13) and the sum over Table 1 with $n_I=1$. For $n_I>1$, the qubit count curves simply shift to the left, as $n_I$ and $N$ only appear together in the qubit scaling, as $N+n_I$.

Figure 5

Gate cost comparison: The dashed line represents the dominant contribution from $U_h$ to the total gate count for QPS with midcircuit measurements.

Figure 6

Probability vs Number of emissions ($E$) for 2, 3, 4, and 5 step simulations. Error bars represent $1\sigma$ ranges, e.g., in each third subplot, the red error bars correspond to the standard deviation of the difference distribution between simulation results obtained from original QPS and QPS with remeasurement. Classical MCMC data were obtained using ${10}^7$ shots, so the statistical errors are suppressed by a factor of $1/10$, and are thus negligible. Error bars in each second subplot are the statistical deviations for $g_{12}=0$ simulations. (a) $N=2$ (b) $N=3$ (c) $N=4$ (d) $N=5$.

Figure 7

Probability vs ${\log }_e({\theta }_{\max })$ for 2, 3, 4, and 5 step simulations. As in Fig. 6, error bars represent $1\sigma$ ranges for the difference distributions. Simulation data (red, blue) is normalized such that probabilities are equal to bar height times bar width. For $g_{12}=0$ simulations (blue), the area of each bar is equal—up to statistical deviations—to the integral of the analytical curve (black) over the respective bin. (a) $N=2$ (b) $N=3$ (c) $N=4$ (d) $N=5$.

Figure 8

Complete illustration of the $U_{\text{count}}$ gate.

Figure 9

Increment gate controlled on $|p=\varphi ⟩$. First, a multi-control gate that encodes whether $|p_i⟩$ is a $\varphi$ onto an ancilla $|c⟩$ is applied. Then, a Ripple Adder [34] with $|c⟩$ as an input carry, and an additional ancillary register initialized to $|0⟩$ is applied. Therefore, $|n_{\varphi }⟩$ is mapped to $|n_{\varphi }+1⟩$ if $c=1$, i.e., if $|p_i⟩$ is a $\varphi$. Increment gates controlled on $|p=f_a⟩$ $|p=f_b⟩$ are identical other than the control setting (see (a2).

Figure 10

A rotation-decrement iterate used to construct $U_h$.

Figure 11

Original $U_h$ gate. Each subgate is defined by Fig. 10.

Figure 12

$U_p$ consists of applying this gate, controlled on $|h_m=j⟩$, for each $j$ from 1 to $n_I+m$.

Figure 13

The $U_h$ gate at step $m$. Each $U_h^{(m,j)}$ gate denotes a sequence of two-level rotations (see Appendix pp1-s8) controlled on the different states stored in $|n_a⟩$ and $|p{⟩}_j$. These controls are denoted by circular “gates” in the diagram. The relative probabilities for an emission from $|p{⟩}_j$ depend on whether $|p{⟩}_j$ is an $f_a$, $f_b$, or $\varphi$, as well as the number of each particle type in the system (which is fully encoded by $|n_a⟩$ and previous measurements of $|h⟩$). Therefore, we must apply $3{\lceil \log }_2(n_I+m+1)\rceil$ different two-level rotations between $|h=0⟩$ and $|j⟩$, each conditional on the values stored in $|n_a⟩$ and $|p{⟩}_j$. Applying a controlled-decrement after each $U_h^{(m,j)}$ ensures that the correct relative emission probabilities are encoded into $|h⟩$ (see Appendix A.5 in [4]), and also resets $|n_a⟩$ to the $|0⟩$ state. Finally, the last gate puts $|e⟩$ back to the $|0⟩$ state—after updating the history register, $|h⟩\ne |0⟩\iff |e⟩=|1⟩$, so we apply a NOT gate conditional on $|h⟩\ne |0⟩$.

Figure 14

$U_p$ gate to be applied if $p_j=\varphi$. Note that prior to the emission, $p_j$ is not encoded on qubits, so the particle update can be though of as “initializing” new registers $|p_j⟩$ and $|p_m⟩$.