Improving the Performance of Deep Quantum Optimization Algorithms with Continuous Gate Sets

Variational quantum algorithms are believed to be promising for solving computationally hard problems on noisy intermediate-scale quantum (NISQ) systems. Gaining computational power from these algorithms critically relies on the mitigation of errors during their execution, which for coherence-limited operations is achievable by reducing the gate count. Here, we demonstrate an improvement of up to a factor of 3 in algorithmic performance for the quantum approximate optimization algorithm (QAOA) as measured by the success probability, by implementing a continuous hardware-efficient gate set using superconducting quantum circuits. This gate set allows us to perform the phase separation step in QAOA with a single physical gate for each pair of qubits instead of decomposing it into two $\mathrm{C}Z$ gates and single-qubit gates. With this reduced number of physical gates, which scales with the number of layers employed in the algorithm, we experimentally investigate the circuit-depth-dependent performance of QAOA applied to exact-cover problem instances mapped onto three and seven qubits, using up to a total of 399 operations and up to nine layers. Our results demonstrate that the use of continuous gate sets may be a key component in extending the impact of near-term quantum computers.

Popular Summary

Quantum computers have the potential to solve problems that today’s computers cannot solve in a reasonable amount of time. However, their computations are not yet reliable, meaning that algorithms with many operations cannot be executed without significant errors. This article presents a method to reduce these errors by reducing the total number of operations required to execute a quantum optimization algorithm. This work thereby offers an approach to solving more complex problems on existing and near-term quantum computers.

The optimization algorithm considered in this work uses an Ising-type interaction between pairs of qubits. In prior work, this interaction was typically realized with a long sequence of standard quantum gates. By developing a gate that directly realizes the desired interaction, this work presents a hardware-efficient implementation that reduces the total number of gates executed on the quantum computer. This reduction in the number of gates results in a lower number of errors and, therefore, improves the overall performance of the algorithm.

The results demonstrate that using hardware-efficient gates is a key component in extending the impact of near-term quantum computers. In the future, the development of related types of hardware-efficient gates might enable quantum computers to tackle an even broader range of problems.

Figure 1

(a) Quantum circuit of a layer $q$ of QAOA for the two-qubit subspace $|Q_i{Q}_j⟩$, using the controlled arbitrary-phase gate (blue) to rotate the |11⟩ state by an angle $2{\mathrm{\Gamma }}_{ij}$ where ${\mathrm{\Gamma }}_{ij}=2{\gamma }_q{J}_{ij}$. (b) A QAOA layer with the phase-separation unitary $U_C^{ij}$ decomposed into $\mathrm{C}Z$ gates (green) and additional Hadamard gates and single-qubit $Z$-gates. (c) Excited-state population $P_{|1⟩}$ of the higher-frequency qubit $Q_i=A_1$ brought in interaction with the lower-frequency qubit $Q_j=B_2$ via a flux pulse (all qubits are operated at their maximal frequency). We perform a two-dimensional sweep of flux-pulse amplitude $a$ and flux-pulse duration $l$, and indicate the maximum population recovery with blue dots. $P_{|1⟩}$ is extracted using a three-level readout (see Appendix pp1). (d) Conditional phase for the dots indicated in (c). The green diamond corresponds to the $\mathrm{C}Z$ gate. The right axis indicates the detuning between |11⟩ and |20⟩. The inset depicts the shift in frequency of the bare |20⟩ state relative to the bare |11⟩ state during the flux pulse for the $\mathrm{C}{Z}_{\varphi }$ gate and the $\mathrm{C}Z$ gate, see Appendix pp2 for more details.

Figure 2

(a) Hardware connectivity graph of the quantum device. Dots correspond to qubits and edges indicate between which pairs of qubits two-qubit gates can be realized. The gray dashed line indicates the subset of qubits used for the three-qubit problem instance depicted in (b). (b) Visual representation of the incidence matrix $K$ (dots indicating entries $K_{\ell i}=1$) for a chosen three-qubit exact-cover problem instance. The labels above the columns (and the colors) indicate which physical qubits are used to represent the corresponding subset. The two solution states are indicated below the grid. (c) Visual representation of the incidence matrix for a chosen seven-qubit problem instance.

Figure 3

Cost function evaluated for $p=1$ on the three-qubit problem instance using $\mathrm{C}{Z}_{\varphi }$ gates. (a) Cost-function landscape as a function of variational parameters. (b) Cost-function landscape obtained from noise-free simulations. (c) Experimental evaluation (blue) and simulation (black) of the cost function for two horizontal line cuts of (a),(b), with ${\beta }^{\prime }=\pi /8$ (dotted lines) and ${\beta }^{″}=3\pi /8$ (dashed lines), respectively. (d),(e) Ten convergence traces of the separation angle and the mixing angle, respectively, for end-to-end optimization starting from random parameter initialization. (f) Average energy (solid black line) and individual convergence traces (faded blue lines) of the energy corresponding to parameters shown in (d),(e).

Figure 4

Output state probability distribution for the three-qubit problem instance implemented with controlled arbitrary phase gates (a)–(c), and decomposed using $\mathrm{C}Z$ gates (d)–(f). States are measured at optimal parameters for depths of $p=1$ (a),(d), $p=2$ (b),(e), and $p=3$ (c),(f). The filled bars correspond to the measured state probabilities in which we highlight the problem solutions in blue (direct implementation) and green (decomposed implementation), respectively. The black wireframes are the expected QAOA outcome from noise-free simulations and the red wireframes are from master-equation simulations. We use 15000 individual measurement runs to estimate each probability distribution. The black vertical markers indicate a bootstrap 99.7% confidence interval [39] for each state, estimated from the bootstrap standard deviation of 100 resampled datasets of the original single-shot measurements.

Figure 5

Performance of QAOA for (a) three qubits and (b) seven qubits. The blue points are implemented with the direct controlled arbitrary-phase gate and the green points are implemented with the gate sequences decomposed into $CZ$ gates. The black squares indicate the highest success probabilities found with noise-free simulations. The top axes indicate the sequence duration for the direct implementation and the decomposed implementation in green and blue, respectively. The gray areas indicate success probabilities above 1. The success probabilities are estimated from 15 000 (three-qubit problem instance) and 60 000 (seven-qubit problem instance) single-shot measurements. The error bars indicate a bootstrap 99.7% confidence interval.

Figure 6

Experimental setup. The experiment is controlled by AWGs whose signals are routed to the quantum device through a series of bandpass filters (BP), lowpass filters (LP), and Eccosorb filters. The flux pulses are combined with a voltage source using a bias-T. The IQ signal from the control AWG is up-converted (UC) to a microwave signal using an IQ mixer. The readout signal is generated by the FPGA SA, and the output from the quantum device is amplified by a chain of amplifiers before being down-converted (DC) and analyzed by the FPGA SA.

Figure 7

Optical micrograph of the device. Each qubit (colors corresponding to the colors in Fig. 2) is connected to neighboring qubits by coupling resonators (white). Each qubit is connected to a readout resonator (red), a flux line (green), and a drive line (pink).

Figure 8

Three-level single-shot readout characterization of qubit $A_1$ with ground-state heralding. We display the first 3000 of the 20 000 single-shot measurements for each state. The blue (red) (green) shaded area correspond to the region of the integrated quadrature plane in which a measured data point is assigned to the $|0⟩$ ($|1⟩$) ($|2⟩$) state. The white dots and dashed white lines correspond to the mean and the $1\sigma$-confidence ellipse of each mode of the mixture model.

Figure 9

Single-qubit error per gate (vertices) and two-qubit error per gate (edges) in percent, measured with randomized benchmarking. We indicate the residual $ZZ$ coupling between each pair of coupled qubits below the corresponding two-qubit gate infidelity. The gate between $A_3$ and $B_2$ is not needed for the problem instances considered in this work (dashed line).

Figure 10

Energy-level diagram and pulse schemes used for the calibration of the $\mathrm{C}{Z}_{\varphi }$ gate. (a) Time evolution of the energy levels of the qutrit pair $|Q_i{Q}_j⟩$ when a flux pulse is applied to $Q_i$ (not drawn to scale). The energy levels directly involved in the gate are shown in black and the interaction between the |11⟩ and |20⟩ states is indicated with two blue arrows. (b)–(d) Pulse scheme used to measure the Chevron pattern, the conditional phase and the dynamic phase, respectively. The $\pi$ pulses, $\pi /2$ pulses, flux pulses, and readout pulses are shown in dark blue, pink, light blue, and purple, respectively. A dashed line indicates that the measurement is repeated once with, and once without the corresponding pulse.

Figure 11

Pulse sequences for implementing QAOA with $p=1$. The shaded area around flux pulses illustrates which interaction they implement and how long buffer times before and after the flux pulse are chosen. (a) Direct implementation of the seven-qubit instance. (b) Decomposed implementation of the seven-qubit instance. (c) Direct implementation of the three-qubit instance. (d) Decomposed implementation of the three-qubit instance.

Figure 12

Simulated and experimental cost-function landscapes of the three-qubit problem instance for $p=1$. The direct implementation is displayed in (a)–(c). The decomposed version is shown in (d)–(f). The master-equation simulations are performed including errors from residual $ZZ$ coupling only (a),(d), and with both residual $ZZ$ coupling and decoherence (b),(e). The experimental data is shown in (c),(f).

Figure 13

Cost function evaluated for $p=1$ on the seven-qubit problem instance using $\mathrm{C}{Z}_{\varphi }$ gates. (a) Cost-function landscape as a function of variational parameters. (b) Cost-function landscape obtained from noise-free simulations. (c) Experimental evaluation (blue) and simulation (black) of the cost function for two horizontal line cuts of (a),(b), with ${\beta }^{\prime }=\pi /8$ (dotted lines) and ${\beta }^{″}=3\pi /8$ (dashed lines), respectively. (d),(e) Ten convergence traces of the separation angle and the mixing angle, respectively, for end-to-end optimization starting from random parameter initialization. (f) Average energy (solid black line) and individual convergence traces (faded blue lines) of the energy corresponding to parameters shown in (d),(e).

Figure 14

Overlay of three selected convergence traces on the cost-function landscape of the (a) three-qubit and (b) seven-qubit problem instance. The random initialization is shown with a black triangle and the final location of each trace is indicated by a round marker whose color reflects the cost of the last function evaluation.

Figure 15

Output state probability distribution for the seven-qubit problem instance implemented with $\mathrm{C}{Z}_{\varphi }$ gates (a) and decomposed using $\mathrm{C}Z$ gates (b). States are measured at optimal parameters for depth of $p=2$ for (a) and $p=1$ for (b). The filled bars correspond to the measured problem solutions, while the black (red) wireframes are the expected QAOA outcome from noise-free (master-equation) simulations. We use 60 000 individual measurement runs to estimate each distribution and indicate bootstrap 99.7% confidence intervals with black vertical markers.