Qubit-ADAPT-VQE: An Adaptive Algorithm for Constructing Hardware-Efficient Ansätze on a Quantum Processor

Quantum simulation, one of the most promising applications of a quantum computer, is currently being explored intensely using the variational quantum eigensolver. The feasibility and performance of this algorithm depend critically on the form of the wave-function ansatz. Recently in Ref. [Nat. Commun. 10, 3007 (2019)], an algorithm termed ADAPT-VQE was introduced to build system-adapted ansätze with substantially fewer variational parameters compared to other approaches. This algorithm relies heavily on a predefined operator pool with which it builds the ansatz. However, Ref. [Nat. Commun. 10, 3007 (2019)] did not provide a prescription for how to select the pool, how many operators it must contain, or whether the resulting ansatz will succeed in converging to the ground state. In addition, the pool used in that work leads to state-preparation circuits that are too deep for a practical application on near-term devices. Here, we address all these key outstanding issues of the algorithm. We present a hardware-efficient variant of ADAPT-VQE that drastically reduces circuit depths using an operator pool that is guaranteed to contain the operators necessary to construct exact ansätze. Moreover, we show that the minimal pool size that achieves this scales linearly with the number of qubits. Through numerical simulations on ${\mathrm{H}}_4$, $\mathrm{Li}\mathrm{H}$ and ${\mathrm{H}}_6$, we show that our algorithm (“qubit-ADAPT”) reduces the circuit depth by an order of magnitude while maintaining the same accuracy as the original ADAPT-VQE. A central result of our approach is that the additional measurement overhead of qubit-ADAPT compared to fixed-ansatz variational algorithms scales only linearly with the number of qubits. Our work provides a crucial step forward in running algorithms on near-term quantum devices.

Popular Summary

Understanding strongly correlated electronic systems is a central goal of condensed-matter physics and quantum chemistry. Achieving accurate simulations of such systems would lead to breakthroughs in fundamental science and in applications, such as medicine and materials science. However, this is computationally intractable even for the best supercomputers. Universal quantum computers would surmount this difficulty, but these are likely a decade or more away. As a result, the community is investigating whether useful simulations can be run on near-term quantum processors. We present a general simulation algorithm that dramatically reduces the performance requirements of the quantum processor, potentially bringing this goal within reach of current technological capabilities.

In this work, we introduce qubit-ADAPT, an algorithm that provides critical advances beyond existing quantum-simulation algorithms. In recent years, it has been recognized that larger, more accurate simulations can be achieved with near-term devices by splitting the work over both quantum and classical processors. Unlike most other algorithms, qubit-ADAPT is designed to adapt itself to the simulation problem, leading to a significant reduction in the demands on the classical processor. At the same time, it also cuts down on the coherence requirements of the quantum processor by utilizing more hardware-efficient logic operations. We demonstrate the power of qubit-ADAPT through numerical simulations of several strongly correlated molecules, finding that the algorithm reduces resource requirements by an order of magnitude while maintaining high accuracy compared to existing methods. This algorithm will enable the study of larger and more complicated systems on near-term quantum processors.

Figure 1

Ground-state energy difference between ADAPT and the exact (FCI) result for (a),(d) ${\mathrm{H}}_4$ at bond distance $1.5\AA$, (b),(e) $\mathrm{Li}\mathrm{H}$ at bond distance $2\AA$, and (c),(f) ${\mathrm{H}}_6$ at bond distance $1.5\AA$. The bond distances are chosen to ensure that correlation effects are substantial. All the calculations are performed using an STO-3G basis and start from restricted Hartree-Fock orbitals without using a frozen-core approximation such that we have eight spin orbitals for ${\mathrm{H}}_4$, 12 spin orbitals for $\mathrm{Li}\mathrm{H}$, and 12 spin orbitals for ${\mathrm{H}}_6$. Results are shown for qubit-ADAPT (blue), fermionic-ADAPT (orange), and transpiled fermionic-ADAPT (green). (a)–(c) Energy difference as a function of ADAPT iterations (which is equal to the number of variational parameters). (d)–(f) Energy difference as a function of the number of cnots in the corresponding state-preparation circuit. All cnot gate counts are obtained using the qiskit command count_ops. The transpiled counts are obtained using qiskit.transpile [47] with heavy optimization, which includes canceling back-to-back cnots and “commutative cancellation” (see Supplementary Material [56] for the code). Although qubit-ADAPT requires more variational parameters, it entails significantly fewer cnots compared to fermionic-ADAPT.

Figure 2

Energy error of qubit-ADAPT versus ansätze with random operator orderings as a function of the number of iterations for (a) ${\mathrm{H}}_4$ and (b) $\mathrm{Li}\mathrm{H}$. The qubit pool is used in all cases, and the orbital bases and bond distances are chosen as in Fig. 1.

Figure 3

qubit-ADAPT and fermionic-ADAPT energy error of $\mathrm{Li}\mathrm{H}$ for various bond distances. The bond distance varies from $1$ to $3\AA$ with darker curves in (a), (b) corresponding to larger bond distance. The energy error is plotted against (a) the number of parameters and (b) the number of cnots. The performance remains similar across different bond lengths and hence across different amounts of correlation in the ground state. The orbital basis used is the same as in Fig. 1. (c) The final energies obtained by qubit-ADAPT (points marked by $\times$) for each bond distance considered are plotted along with the exact ground-state energies (solid curve).

Figure 4

qubit-ADAPT energy error of ${\mathrm{H}}_4$ using the full qubit pool, and randomly chosen 1/4, 1/16, and 1/32 pools are plotted against the number of iterations. For the 1/4 and 1/16 pools, the algorithm can converge for almost every run. For the 1/32 pool, most of the runs can reach only an energy error of ${10}^{-3}$. The orbital basis and bond distance are chosen as in Fig. 1.

Figure 5

The fraction of pools that are complete as a function of pool size for two (blue circles), three (yellow squares), four (green diamonds), and five (red triangles) qubits.

Figure 6

Energy error curves for three different kinds of minimal complete pools versus incomplete pools. Results for three, four, and five-qubit random Hamiltonians are shown in (a), (b), and (c), respectively.