Hybrid Quantum-Classical Eigensolver Without Variation or Parametric Gates

Quantum computers offer the ability to address problems in quantum many-body chemistry and physics by quantum simulation or in a hybrid quantum-classical approach. The latter method is considered the most promising approach for noisy-intermediate scale quantum (NISQ) devices Preskill (2018). The prospect and benefits of quantum algorithms, along with suitable hardware, is in overcoming the complexity of the wave-function of a quantum system as it scales exponentially with system size McArdle et al. (2020). Therefore developing techniques and algorithms for NISQ era devices that may prove to have some computational advantage themselves, or establish a path towards ideas and foundations that provide advantage for future error-corrected quantum devices, is a worthwhile pursuit.

The leading algorithms intended to be executed on NISQ devices, which aim to determine solutions to an electronic Hamiltonian, are variational in nature Yuan et al. (2019). One specific algorithm is the variational quantum eigensolver (VQE), which has been tremendously successful in addressing chemistry and physics problems on quantum hardware and NISQ devices Peruzzo et al. (2014); McClean et al. (2016); Kandala et al. (2017); Barkoutsos et al. (2018); Jones et al. (2019); Nakanishi et al. (2019); Parrish et al. (2019a); Higgott et al. (2019); Jouzdani et al. (2019). However, the restriction or challenge that exist with VQE is the need for prior insight with regard to selecting the trial quantum state, i.e., ansatz circuit. Furthermore, the classical optimization of the ansatz parameters may be a poorly converging problem McClean et al. (2018); Parrish et al. (2019b) and therefore limiting the applicability of VQE for obtaining results accurate enough for chemical or physical interpretation. Finally, the realization of ansatz circuits that are motivated by domain knowledge, for example the unitary coupled cluster ansatz for chemistry problems Romero et al. (2017), may not be directly applicable on NISQ hardware and therefore requires clever modification to obtain hardware efficient ansätze Kandala et al. (2017); Barkoutsos et al. (2018); Herasymenko and O’Brien (2019).

In this work, we present a pragmatic hybrid quantum-classical approach for calculating the eigenenergy spectrum of a quantum system within an effective model. Firstly, an effective Hamiltonian is obtained through measurement of short-depth quantum circuits. The effective Hamiltonian is essentially the projection of the quantum system Hamiltonian onto a limited set of computational bases. The basis set is prepared with the intent of ensuring the dimensions of the corresponding matrix does not grow exponentially with the system size. In order to evaluate the matrix elements of the effective Hamiltonian, suitable non-parametric quantum circuits are specified. The quantum circuits are designed, executed, and measured. From the result of the measurements, the diagonal and off-diagonal terms of the effective Hamiltonian matrix are obtained. On the classical side, the effective Hamiltonian matrix, with suitable dimensions, is diagonalized numerically using a classical computer.

The paper is organized as follows: A short background is presented in Section II. In Section III the steps taken in our hybrid quantum-classical approach are explained in detail. In Section IV we demonstrate the application of this hybrid approach on simple chemical molecules BeH₂ and LiH. In Section V the approach is demonstrated on the IBMQ 5-qubit Valencia quantum processor IBM-Q team (2020a) for H₂ molecule. Finally, we discuss the integration of VQE and the proposed approach in Section VI.

Consider a quantum many-body system of electrons with the second quantized Hamiltonian:

$a_{i}^{\dagger}$ and $a_{i}$ are the creation and annihilation operators, respectively. The anticommutator for the creation and annihilation are given by: $a_{i}a^{\dagger}_{j}+a^{\dagger}_{j}a_{i}=\delta_{ij}$ and $a_{i}a_{j}+a_{j}a_{i}=a^{\dagger}_{i}a^{\dagger}_{j}+a^{\dagger}_{j}a^{\dagger}_{i}=0$. These rules enforce the non-abelian group statistics for fermions, that is, under exchange of two fermions the wave-function yields a minus sign.

The indices in Eq. (1) refer to single-electron states. The coefficients $\kappa_{ij}$ and $v_{ijkl}$ are the matrix integrals

and

where $\hat{K}_{1}$ and $\hat{V}_{12}$ operators correspond to one- and two-body interactions respectively. Since $\hat{K}_{1}$ and $\hat{V}_{12}$ can depend on other parameters, such as the distance between nuclei, the Hamiltonian in Eq. (1) represents a class of problems. However, this class of problems has the common property that the number of fermions is a conserved value. Strictly speaking, the terms in the Hamiltonian act on fixed-particle-number Hilbert spaces, $\mathcal{H}_{N_{F}}$, that have the correct fermionic antisymmetry, with $N_{F}$ denoting the number of electrons. In this paper, we consider this class of electronic systems where the Hamiltonian is assumed to be in the form of Eq. (1). The coefficients expressed in Eqs. (2–3) can be obtained using software packages developed for quantum chemistry calculations that perform efficient numerical integration Reine et al. (2012).

The application of the methodology discussed in Section III is focused on BeH₂ and LiH molecules due to thier relatively small number of electrons and molecular orbital footprint. The number of electrons in BeH₂ and LiH is six and four, respectively. The single-electron molecular spin-orbitals in the second quantized Hamiltonian are constructed using the minimal atomic STO-3G basis set Kandala et al. (2017); McArdle et al. (2020). For BeH₂ there are a total of $14$ spin-orbital, and thus corresponds to $14$ qubits; LiH contains $12$ spin orbitals and hence $12$ qubits. For the purpose of measuring the matrix elements of the effective Hamiltonian, Eq. (7), one additional ancillary qubit is required, as illustrated in the quantum circuits shown in Fig. 1. Therefore, the total number of qubits for the simulation of BeH₂ and LiH is $15$ and $13$, respectively.

To obtain the coefficients in the Hamiltonian of Eq. (1) — more specifically as defined in Eqs. (2) and (3) — we make use of the Psi4 quantum chemistry package Parrish et al. (2017). The second quantized Hamiltonian is further transfomed onto a set of Pauli strings and their corresponding weights, Eq. (1), via JW transform using OpenFermion package McClean et al. (2017a).

In order to construct the potential energy surface for each inter-nuclear distance, $R$, the steps indicated in the previous paragraph are repeated. The distance $R$ corresponds to the bond length between Be–H or Li–H in a given molecule; both LiH and BeH₂ are linear molecules. We note that these calculations are assuming the total Hamiltonian can be represented using the Born-Oppenheimer approximation, where the dynamics of the core nuclei are neglected. This is standard practice in quantum chemistry calculations Jensen (2017) and thus at every distance $R$, the nuclear-nuclear repulsion energy is treated classically and is added to the Hamiltonian as a constant.

For every distance $R$, a set of computational bases is chosen. This set includes the basis with the lowest energy expectation (the reference configuration) and computational bases that correspond to the low-order excitations (i.e., S, D, etc.) with respect to the reference configuration. In the case of BeH₂, we construct the effective Hamiltonian matrix by keeping the S, D, and T excitations, and thus the total number of computational bases for BeH₂ is $N_{s}=1588$. The low dimension of the effective Hamiltonian matrix makes it possible to diagonalize the matrix on a classical computer using standard numerical techniques Virtanen et al. (2020). Thus, the lowest energies, including the ground state energy, are obtained at every $R$. The results for this process are shown in Fig. 3.

Fig. 3(a) shows the ground state energy as well as a few low-lying excited states for BeH₂ of the effective Hamiltonian. The exact energies are given by the dashed curves in Fig. 3(a). In Fig. 3(b), the difference between the exact and obtained ground state energy is shown. An error within the hatched area indicates results that are within chemical accuracy. Chemical accuracy is typically identified as $\sim 5\times 10^{-3}$ Hatrees. Figure 3(c) shows the same energy difference for all other excited-state energies. The same process is done for LiH where only S and D configurations are used. The total number of permutations of $4$ fermions ($495$) is reduced to $N_{s}=200$, and the results are shown in Fig. 4.

The total number of configurations can be approximated by the highest excitation considered. For $n$-electron excitation the number of configurations is $\binom{N_{F}}{n}\binom{N-N_{F}}{n}<N^{2n}$. Thus, the total number of configurations is less than $1+\cdots+N^{2n}=\mathcal{O}(N^{2n})$ when $N\gg 1$. Of course, this argument is valid as long as $n$ can be assumed to be finite and independent of the size of the system $N$ or the number of electrons $N_{F}$. Under these assumptions, the dimensions of the effective Hamiltonian, $H_{eff}$, remains polynomial if one replaces the minimal STO-3G with the extended atomic basis.

In this section, we demonstrate the feasibility of the hybrid quantum-classical approach discussed in this paper on a IBMQ hardware device. Specifically, we calculate the eigenenergy spectrum of H₂ molecule using STO-3G minimal basis-set.

Within STO-3G basis set, the total number of spin-orbitals for H₂ is four. In contrast to the numerical demonstration, where the computational basis was prepared using the UCC ansatz, here we employ the complete computational basis with conserved number of electrons for H₂ ($N_{F}=2$). The number of computational basis set with conserved electron number, is $N_{s}=6$, and the bases are:

The number of bases is not further reduced; that is, the effective Hamiltonian constructed here is equal to the full Hamiltonian within the $N_{F}=2$ electron number. The above set can be interpreted as the collection of all the S and D exciations, as introduced in sub-section III.3, above a classical configuration such as $\ket{1100}$.

The Hamiltonian of the problem, stated as a weighted sum of Pauli string operators, as in Eq. (4), has a total of fifteen operators, which can be grouped as:

and

Here, a Pauli string operator such as $ZIII$ is a shorthand for $\hat{Z}_{1}\otimes\hat{I}_{2}\otimes\hat{I}_{3}\otimes\hat{I}_{4}$, as introduced in Eq. (5).

Our goal is to use the quantum hardware to evaluate the values for $\bra{\bf n}\hat{h}_{s}\ket{\bf n^{\prime}}$, where $\hat{h}_{s}$ belongs to $G_{1}$ or $G_{2}$, and $\ket{\bf n}$ and $\ket{\bf n^{\prime}}$ can be any of the bases listed in Eq. (19).

Note that this task is not a difficult one for a classical computer. However, the objective of the paper and hardware demonstration is to establish a hybrid quantum-classical computational scheme in which the quantum processor performs some of the computational tasks. In the simulations of this paper, the task is the evaluation of the phase associated with the exchange of fermions; since $\bra{\bf n}\hat{h}_{s}\ket{\bf n^{\prime}}$ can be understood as exchanging fermions on $\ket{\bf n}$ via the operator $\hat{h}_{s}$, and to arrive at $\ket{\bf n^{\prime}}$. The output can only take discrete values of $\pm 1$, $\pm i$, and $0$.

In each group $G_{1}$ and $G_{2}$, Pauli string operators commute and thus, theoretically, can be measured simultaneously. However, in order to perform simultaneous measurement, one needs to adjust the computational bases, by applying further quantum gates on the target qubits. The appended measurement circuit further increases the circuit depth Gui et al. (2020). Thus, practically, the simultaneous measurement may not be advantageous, and, in contrast, induces further noise and decoherence on NISQ devices. Nevertheless, for $G_{1}$, no further circuit depth is required, as they are all measured in the $z$-basis.

We can further reduce the quantum coprocessor computational load by excluding circuits containing terms that are classically efficient to compute. In eqs. (20-21), the operators in $G_{1}$ do not contribute to off-diagonal matrix element of the Hamiltonian, while operators in $G_{2}$ contribute only to the off-diagonal matrix elements. This information is used to reduce the essential number of circuits to be run on the hardware.

The evaluation of an element $\bra{\bf n}\hat{h}_{s}\ket{\bf n^{\prime}}$ is performed by assigning an appropriate circuit, as introduced in sub-section III.1, to a quantum coprocessor. In this work we make use of the IBMQ cloud open devices.

Once a circuit is loaded onto a quantum coprocessor, each circuit is executed a several times. The number of times a circuit is run is referred to as the number of shots. In this work, each circuit was executed for a total of $8,000$ shots. At each run, the qubits are measured, and the collection of the realized configurations is used to evaluate value of $\bra{\bf n}\hat{h}_{s}\ket{\bf n^{\prime}}$.

The preparation of the circuits is performed using the transpile and assemble routines available in the IBM Qiskit API Abraham et al. (2019), which enables collating individual circuits, corresponding to the diagonal and off-diagonal terms, to prepare each circuit to run as a single job on a target IBMQ device. In this work, we make use of the IBMQ 5-qubit Valencia hardware IBM-Q team (2020a). The number of circuits — for both real and imaginary parts separately — after the transpile and assemble routines for the diagonal and off-diagonal terms corresponds to 66 and 60 circuits, respectively. The execution timings for the IBMQ QASM simulator range between 0.21-1.5 seconds and for the IBMQ Valencia 5 qubit device 274-298 seconds with 8000 shots per circuit. Due to the limited number of qubits available on this processor, direct measurement is used, that is by measuring all the qubits at the end of the circuit.

As a sanity check, we first use IBMQ QASM simulator IBM-Q team (2020b) to perform all steps involved as outlined in III. The calculations are shown in Fig. 6(a). The results perfectly match the exact values as expected. Furthermore, in Fig. 6(b) the off-diagonal terms in the Hamiltonian are evaluated using IBMQ Valencia device, while the diagonal elements are evaluated using the IBMQ QASM simulator. In both cases, upon diagonalization of the obtained Hamiltonian, the energy spectrum is in good agreement with the exact values. We observe in Fig. 6(b) that there is a slight lifting of the double degeneracy of H₂, which we associate to the device noise. This is due to the occurance of non-zero values in the measured off-diagonal terms when in fact the exact value should be zero.

Finally, all the matrix elements, that is, diagonal and off-diagonal elements, are evaluated using IBMQ Valencia device. The results are shown in Fig. 7-(a), which was done with and without measurement error mitigation. The use of measurement error mitigation marginally improves the results. The discrepancy between the exact and hardware values, without and with measurement error mitigation, are shown in 7(b-c).

The effect of noise is ubiquitous in current quantum hardware and is inherently complex and difficult to characterize. However, the result of noise due to the measurement process can be mitigated to some degree by using a calibration matrix Abraham et al. (2019). The resulting calibration matrix is then used to reduce errors of subsequent circuit measurements. The application of measurement error mitigation is demonstrated in Fig. 7(a) with beneficial impact on error shown in Fig. 7(c).

The calculations in Fig. 7, that is construction of the effective Hamiltonian, $\hat{H}_{eff}$, are performed through five independent IBMQ job submissions. Each time the energy-surface is slightly different, which can be associated to the inherent device errors and perhaps low number of shots used. The error bars in Fig. 7(b-c) indicate the min/max range in errors of the five different calculation.

Recently an abundance of Variational Quantum Algorithms (VQA) have been introduced to solve electronic quantum many-body problems on NISQ devices Peruzzo et al. (2014); Wecker et al. (2015); McClean et al. (2016); Yuan et al. (2019); Parrish et al. (2019a); Higgott et al. (2019); Jones et al. (2019); Nakanishi et al. (2019). These VQA proposals mostly rely on optimization of parametric circuits. In this paper we demonstrate that an alternative approach exists, specifically for quantum chemistry problems, which does not require an optimization procedure and parametric ansatz. In this approach, the quantum hardware is used only to prepare a many-body quantum state and to efficiently measure the expectation values of certain target observables with respect to this quantum state. From the output measurements, one can construct what we refer to as an effective Hamiltonian matrix, $\hat{H}_{eff}$. Upon diagonalization of $\hat{H}_{eff}$ using classical eigen-decomposition numerical methods one obtains the ground state and low-lying excited states of the system.

The approach introduced in this paper is particularly suitable for NISQ devices where short quantum circuit depth is essential due to lack of error-correction protocols on these devices. An additional important aspect of this approach is that it provides access to the low-lying energy spectra of the system and not just the ground state in comparison with the original VQE process.

In the context of VQE and VQE-type algorithms, several attempts have been made to extend the variational approach to excited states Tilly et al. (2020); Santagati et al. (2018); Higgott et al. (2019); McArdle et al. (2019); Somma (2019); Parrish et al. (2019a). Quantum subspace expansion McClean et al. (2017b); Colless et al. (2018) for example constructs a set of non-orthogonal bases out of an optimized ansatz, and performs post-processing to obtain excited states. Deflation techniques as described in ref. Higgott et al. (2019), constructs a pseudo-Hamiltonian in which the ground state is excluded and orthogonality is enforced through regularization. Successful examples are introduced for some low-lying excited states of LiH Jones et al. (2019). In all these previous works, optimization of a parametric ansatz is required therefore necessitates an enormous number of quantum circuit executions and sampling.

The approach introduced in this paper is similar to multistate contracted VQE (MC-VQE) in ref. Parrish et al. (2019a). The main difference is the application of VQE: MC-VQE is obviously a VQE-type algorithm, our approach is distinct in that it does not require a parametric circuit or variational procedure for optimization. In addition, our method differs since it uses supporting quantum circuit resources, i.e., ancilla qubits, in order to perform interference and measurement that is different from the quantum circuit in ref. Diker (2016) and its generalization in ref. Parrish et al. (2019a).

An extension of our approach to VQE type algorithm is possible. This can be done by appending a set of parametric gates that act only on the target qubits to the circuits in Fig. 1. Let us denote this part of the circuit with $U(\theta)$, where $\theta$ stands for a set of parameters. Then, it can be verified that, given $\theta$, the final matrix element obtained from the circuit after measurement (see Eq. (2) for example) becomes $\bra{\bf n}U^{\dagger}(\theta)\hat{H}U(\theta)\ket{\bf n^{\prime}}$, compared to $\bra{\bf n}\hat{H}\ket{\bf n^{\prime}}$. The appended parametric circuit $U(\theta)$ allows one to project the Hamiltonian onto $\mathcal{S}(\theta)=\{U(\theta)\ket{\bf n}\}$, for a given $\theta$. This means the $N_{s}\times N_{s}$ effective Hamiltonian is now parametric and depends on the value(s) of $\theta$. The optimal parameter(s) are then obtained by minimizing the ground-state energy of the effective Hamiltonian matrix. The reference state $\{U(\theta)\ket{\bf n}\}$ can be regarded as the contracted reference states introduced in ref. Parrish et al. (2019a).

One possible limitation of the circuits shown in Fig. 1 is the execution of two-qubit gates corresponding to control operations over $n$-qubits (i.e., series of CNOT gates). For NISQ devices with hardware-restricted qubit connectivity, this may require a number of SWAP gate operations and therefore can increase the circuit depth and subsequent error rates Venturelli et al. (2018); Nash et al. (2020). In essence, the implementation of the circuits described in this paper will depend on the ability to limit circuit depth and associated error rates by NISQ hardware circuit optimization (i.e., scheduling). However, significant improvements in qubit connectivity of various modalities (e.g., ion traps) Wright et al. (2019); Hazra et al. (2020) or optimizing quantum circuits against decoherence Zhang et al. (2019); Holmes et al. (2020) may blunt this concern.

The ideas and methodology discussed in Section III have been supported by General Atomics internal R&D funds. Their application towards quantum chemistry problems as discussed in Section IV and Section V is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, under Award Number DE-SC0020249. We thank Mark Kostuk for his guidance and management as PI under this grant. Additionally, we thank David P. Schissel for feedback and guidance at General Atomics.

Circuit diagrams are rendered using the LaTeX Quantikz package Kay (2019), numerical scripts utilize the Python SciPy package Virtanen et al. (2020), and 2D plots are generated with the Python matplotlib package Hunter (2007). We acknowledge use of the IBMQ for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBMQ team.

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