Comparing planar quantum computing platforms at the quantum speed limit

An important aspect that strongly impacts the experimental feasibility of quantum circuits is the ratio of gate times and typical error time scales. Algorithms with circuit depths that significantly exceed the error time scales will result in faulty quantum states and error correction is inevitable. We present a comparison of the theoretical minimal gate time, i.e., the quantum speed limit (QSL), for realistic two- and multi-qubit gate implementations in neutral atoms and superconducting qubits. Subsequent to finding the QSLs for individual gates by means of optimal control theory we use them to quantify the circuit QSL of the quantum Fourier transform and the quantum approximate optimization algorithm. In particular, we analyze these quantum algorithms in terms of circuit run times and gate counts both in the standard gate model and the parity mapping. We find that neutral atom and superconducting qubit platforms show comparable weighted circuit QSLs with respect to the system size.

Figure 1

Comparison of the circuit run times and gate counts for the quantum Fourier transform (QFT) and quantum approximate optimization algorithm (QAOA) between neutral atoms and superconducting circuits. The left (right) column corresponds to the QFT (QAOA) and the orange (purple) marker correspond to neutral atoms (superconducting circuits). The circuit run times for both algorithms and various problem instances of different size, $N=9,16,\cdots ,81$ qubits for the QFT and $N=9,16,\cdots ,121$ qubits for the QAOA, are given in panels (a) and (c), respectively. The numbers of two- and multi-qubit gates in the corresponding circuits are given in panels (b) and (d), respectively. The data for the squares [circles] is obtained using the standard gate model (SGM) using the gates and times from the standard gate set (SGS) [QSL gate set (QGS)], cf. Table 1. In contrast, the plus signs [crosses] represent the results when the same algorithms are realized in the parity mapping (PM) using the SGS [QGS]. The pentagons correspond to an altered 2D architectures allowing for next-to-nearest-neighbor (NNN) coupling between qubits. The run time for each circuit is given in units of typical platform-specific error times. The gray area in panels (a) and (c) indicates where the circuit run times exceeds the error times.

Figure 2

Reduction in weighted circuit run times and (two- and multi-qubit) gate counts for the scenarios and data shown in Fig. 1. In panels (a) and (b) the circuits in the standard gate model (SGM) and parity mapping (PM) are compared when using the “QSL gate set” (QGS) instead of the “standard gate set” (SGS). In panels (c) and (d) the circuits in the SGS and QGS are compared when using the PM instead of the SGM. All presented numbers are the average over problem instances of different sizes, i.e., different numbers of qubits.

Figure 3

Overview of the QSLs for the various gates of the “QSL gate set” (QGS) in Table 1. The left (right) column shows the results for neutral atoms (superconducting circuits) for three different configurations of control fields (specified in the main text), respectively. Each marker represents the gate error ${\varepsilon }_T$, cf. Eq. (7), after either reaching ${\varepsilon }_T\le {\varepsilon }_{\max }={10}^{-3}$ or 1500 iterations of Krotov's method, cf. Appendix pp2, after starting from a set of random guess fields generated via Eq. (C1). While the lines connect the lowest errors reached for each gate time $T$ within a given field configuration, the shaded background color indicates the range between the lowest and highest error. The marker density is shown at the right side of each panel as a histogram. The parameters for neutral atoms are $V/2\pi =40\mathrm{M}\mathrm{Hz}, 0\le {\mathrm{\Omega }}_{\downarrow }\left(t\right),{\mathrm{\Omega }}_{\uparrow }\left(t\right)\le {\mathrm{\Omega }}_{\max }=0.1V$ and $\left|\mathrm{\Delta }\left(t\right)\right|\le {\mathrm{\Delta }}_{\max }=0.3V$. The parameters for superconducting circuits are $-40\mathrm{M}\mathrm{Hz}\le g_{nm}\left(t\right)/2\pi \le 5\mathrm{M}\mathrm{Hz}, 6700\mathrm{M}\mathrm{Hz}\lesssim {\omega }_n\left(t\right)\lesssim 7100\mathrm{M}\mathrm{Hz}$ (exact values depending on $n$ and taken from Ref. [8]) and $-50\mathrm{M}\mathrm{Hz}\le {\overline{\mathrm{\Omega }}}_{n,\mathrm{re}}\left(t\right),{\overline{\mathrm{\Omega }}}_{n,\mathrm{im}}\left(t\right)\le 50\mathrm{M}\mathrm{Hz}$. The anharmonic ladder for each transmon has been truncated after five levels with population in the highest level suppressed during optimization. The optimization results for $\mathrm{ZZZ}\left(\gamma \right)$ and $\mathrm{ZZZZ}\left(\gamma \right)$, cf. Eq. (A6), have been obtained for the maximally entangling gate at $\gamma =\pi /4$. The results for the CNOT gate on neutral atoms have been obtained using site-dependent control fields for each atom.

Figure 4

Panel (a) shows the entanglement power [97] of the three- and four-qubit constraint gates $\mathrm{ZZZ}\left(\gamma \right)$ and $\mathrm{ZZZZ}\left(\gamma \right)$, cf. Eq. (A6), as a function of $\gamma$. In contrast, panels (b) and (c) examine the QSLs for these gates under various conditions on neutral atoms. In panel (b), the dependence of the QSL on the parameters $\gamma$ is shown. In panel (c), the impact of ${\mathrm{\Delta }}_{\max }/{\mathrm{\Omega }}_{\max }$ is visualized. In the latter case, we have $\gamma =\pi /4$ and a fixed ${\mathrm{\Omega }}_{\max }$ while ${\mathrm{\Delta }}_{\max }$ is modified. The QSLs in panels (b) and (c) have been determined using the parameters and “phase” configuration described in Fig. 3.

Figure 5

Overview of different QSLs similar to Fig. 3 but exclusively for the three- and four-qubit constraint gates $\mathrm{ZZZ}\left(\gamma \right)$ and $\mathrm{ZZZZ}\left(\gamma \right)$, cf. Eq. (A6). The left column shows results for neutral atoms, obtained using the “phase” configuration of Fig. 3, for the pseudo-2D architecture (triangles) and an actual, planar 2D architecture (stars). The right column compares the results for superconducting circuits using the “no-X” configuration from Fig. 3. The data correspond to the physical architecture of Ref. [8] where only nearest-neighbor (NN) couplings between qubits are present (triangles) and where diagonal couplings, i.e., next-to-nearest-neighbor (NNN) couplings, are added (stars).

Figure 6

Illustrations of the quantum circuits for a QFT (a) and a single QAOA step (b) for $N=4$ qubits, respectively.