# Making fermionic quantum simulators more affordable

This is a Perspective on "Resource-Optimized Fermionic Local-Hamiltonian Simulation on a Quantum Computer for Quantum Chemistry" by Qingfeng Wang, Ming Li, Christopher Monroe, and Yunseong Nam, published in Quantum 5, 509 (2021).

By Daniel Leykam (Centre for Quantum Technologies, National University of Singapore).

One of the most promising applications of future quantum computers is the ab-initio calculation of chemical properties including ground state energies [1]. The required number of qubits and gates scales more favourably with the size of the basis set used (linear and polynomial, respectively) compared to the exponentially-growing resources required to merely store a wave function on a classical computer.

Despite the better scaling of quantum algorithms, interesting classically intractable quantum chemistry problems remain well out of reach of near-term quantum computers due to the huge prefactors and overheads involved [2]. Thus, the reduction of resource requirements of quantum algorithms for quantum chemistry is an important problem attracting great interest [3,4,5,6,7,8].

The study by Wang et al. [9], recently published in $\textit{Quantum}$, develops schemes to reduce the number of expensive quantum gates required to compute the ground state energies of fermionic systems with local interactions using quantum computers. They present methods applicable to both near-term quantum processors and eventual fault-tolerant quantum computers.

First, the authors consider the calculation of ground state energies on future fault-tolerant quantum computers, based on applying the quantum phase-estimation algorithm to the time-evolution operator $U = \exp(-i Ht)$ generated by the system’s Hamiltonian $H$. Implementing the time-evolution operator on the quantum circuit is the main challenge for quantum phase estimation, but several efficient methods have been proposed [6]. For example, the Taylor-series approach decomposes $\hat{U}$ into a sum of powers of $H$ obtained via measurement of ancilla qubits [10], while the asymptotically optimal quantum signal processing employs controlled rotations of ancilla qubits [11].

Here the authors consider the more qubit-efficient product-formula method, which splits $\hat{U}$ into a product of simpler unitaries [12,6]. They show that the two-body interaction terms arising from the product-formula decomposition can be recast as triply-controlled rotations requiring fewer costly ancilla qubits and $T$ (phase shift) gates compared to previous optimized implementations. For example, the method of Ref. [13] requires twice as many $T$ gates and eleven times as many ancillas, making this an impressive reduction of the required resources.

Next, the authors consider the variational quantum eigensolver, which is one of the most promising algorithms for near-term quantum processors due to its lower circuit depth and resilience to systematic gate errors [14]. The variational quantum eigensolver is a hybrid quantum-classical algorithm, which uses a parameterized quantum circuit to prepare a trial state and then measures the expectation value of the Hamiltonian operator $H$. The circuit parameters are iteratively optimized in order to minimize the energy and approximate the ground state of $H$.

Obtaining an accurate solution using the variational quantum eigensolver requires an encoding circuit that can well-approximate the ground state of $H$. Here the challenge is to balance the competing requirements of having a low circuit depth, a sufficiently expressive circuit, and being able to efficiently optimize the circuit parameters. For example, hardware-efficient encoding circuits formed by sequences of native gates minimize the circuit depth [3], but suffer from vanishing gradients of the energy with respect to the circuit parameters [15].

Problem-inspired strategies such as the unitary coupled-cluster method attempt to construct trial solutions by optimizing excitations about some reference states [4,7,8]. However, the excitation operators need to be decomposed into sequences of native gates, resulting in deeper circuits. One strategy for minimizing the required circuit depth is to iteratively build up the ansatz circuit, adding parameterized excitations until convergence is achieved [8].

Wang et al. incorporate perturbation theory to the iterative construction of a unitary coupled-cluster ansatz. First, a pool of excitation terms is optimized. Next, second order perturbation theory is used to estimate the influence of additional excitation terms. This allows one to not only identify the best excitation to add to the ansatz, but also estimate how much it will reduce the energy, while only incurring a small overhead of additional observables to be measured. Building up the ansatz in this manner simplifies the classical optimization of the variational parameters while minimizing the circuit depth.

As a second optimization of the resources required for the variational quantum algorithm, Wang et al. study generalized fermion-to-qubit transformations with the aim of reducing the number of costly two-qubit gates required to implement the unitary coupled-cluster ansatz. While numerically finding the global optimal transformation for a given problem is extremely challenging, heuristic methods applied to small molecules including H$_2$O indicate potential savings of up to 20% fewer two-qubit gates. This provides an alternative to generalized transformation approaches which reduce the required number of qubits to encode the wave function at the expense of increasing the circuit depth [5,16].

By reducing the resources required for simulation of fermionic systems on quantum computers, Wang et al. bring us a step closer to demonstrating a quantum advantage for quantum chemistry problems. In the near future, it will be interesting to implement these optimizations to the variational quantum eigensolver on real hardware in order to test their robustness to device noise and the imperfect estimation of observables using finite numbers of measurements. Another pressing issue is to test how well these methods scale as the number of orbitals is increased to the level required to achieve chemical accuracy.

### ► References

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