Variational quantum eigensolvers by variance minimization

The variational principle is instrumental for understanding physical theories and also becomes a powerful tool for solving computational physics problems. In recent years, the power of quantum computing of near-term noisy quantum processors is expected to be exploited with variational methods Yung et al. (2014); Farhi et al. (2014); McClean et al. (2016); O’Malley et al. (2016); Li and Benjamin (2017); McClean et al. (2017); Shen et al. (2017); Kandala et al. (2017); Hempel et al. (2018); Anschuetz et al. (2018); Mitarai et al. (2018); Moll et al. (2018); Endo et al. (2018); Kokail et al. (2019); Takeshita et al. (2020); McArdle et al. (2019); Higgott et al. (2019); Wu and Hsieh (2019); Verdon et al. (2019); Liu et al. (2019a, b); Benedetti et al. (2019); Grimsley et al. (2019); Chowdhury et al. (2020); Wang et al. (2020); Zhang and Yin (2020), which refers to hybrid quantum-classical algorithms for optimization. A representative example is variational quantum eigensolver (VQE) Yung et al. (2014); McClean et al. (2016); O’Malley et al. (2016); Kandala et al. (2017); McArdle et al. (2019); Liu et al. (2019a); Grimsley et al. (2019), aiming to solve eigenstates for a quantum system that receives special interests due to its fundamental roles in quantum chemistry Yung et al. (2014); McClean et al. (2016); O’Malley et al. (2016); Kandala et al. (2017); Grimsley et al. (2019); Arute et al. (2020), quantum many-body systems Liu et al. (2019a); Kokail et al. (2019); Dallaire-Demers et al. (2020), and many other applications Anschuetz et al. (2018); Xu et al. (2019); Lubasch et al. (2020).

Interestingly, while called as eigensolver, VQE typically solve the ground state of a Hamiltonian, using the energy as a cost function to optimize. From the variational principle, such a cost function can gives an upper bound to the ground state energy. For being an eigensolver, many variants of VQE have been developed to solve excited states, which can be understood as minimizing energy in a subspace, constrained by symmetries Greene-Diniz and Ramo (2019), enforced by orthogonality Nakanishi et al. (2019), or obtained by eliminating the space of lower energy states Higgott et al. (2019), or using quantum subspace expansion McClean et al. (2017).

While those variants of VQE can be successful to some extent, an alternative approach is to directly access whether a state is an eigenstate and use this criteria as a cost function to optimize. A simple answer for this is to use zero energy variance to justify an eigenstate since an eigenstate should have zero energy variance. Variational methods based on the zero-energy variance principle Bartlett et al. (1935); Siringo and Marotta (2005); Umrigar et al. (1988); Umrigar and Filippi (2005); Khemani et al. (2016); Pollmann et al. (2016); Vicentini et al. (2019), in fact, can be originated to the early age of quantum mechanics for solving the Helium atom Bartlett et al. (1935). Remarkably, zero variance puts a very strong constraint of the wavefunction structure, and it can be powerful for solving quantum many-body problems when well-approximated wavefunction is required Pollmann et al. (2016). The variance principal has been applied experimentally to self-verify if a quantum simulator can prepare a ground state for a Hamiltonian Kokail et al. (2019). It is, however, awaiting for exploiting the power of quantum computing to develop a variational quantum eigensolver that is based on minimizing energy variance.

In this paper, we develop a variational quantum eigensolver based on minimizing energy variance, which we call as variance-VQE. For solving excited states of a Hamiltonian, we give two approaches, which represent eigenstates in different ways. One uses a single wavefunction ansatz, where different parameters correspond to different eigenstates. The other uses a set of orthogonal wavefunction ansatz, and incorporates energy variances into one cost function to optimize. We numerically solve the energy potential surface for molecules, which is a fundamental quantum chemistry problem, and demonstrate properties and advantages of variance VQE. We also show that optimizing a combination of energy and variance may be more efficient in finding a set of eigenstates with lowest energies, compared with one that optimizes energy or variance alone. Moreover, we also investigate stochastic gradient descent for optimizing VVQE with Hamiltonian sampling, which can be useful for optimization with less quantum resources. Our work demonstrates that VQE by minimizing energy variance can be useful for calculating excited states, and also work out an avenue for efficient optimization.

The paper is organized as follows. We first propose variance VQE in Sec. II, and then apply it to solve excited states for molecules in Sec. III. In Sec. IV, we investigate the stochastic gradient for optimizing VVQE. Finally, we give a summary in Sec. V.

In this section, we first review the traditional approach of variational quantum eigensolver that minimizes the energy. Then, we formulate a new type of VQE that minimizes the energy variance and then discuss its optimization.

As all eigenstates always have zero variance, optimizing variance-VQE by minimizing energy variance may lead to any eigenstate. Thus, variance-VQE can be used for obtaining excited states. In this section, we propose different approaches for solving excited states of molecule Hamiltonians with variance VQE. We first show that the cost function variance-VQE can have many global minimums of zero variance, corresponding to different excited states. However, this approach can be inefficient if a specified eigenstate is required. To overcome this, we propose variance-VQE that solves a set of excited states jointly, using a set of orthogonal wavefunction ansatz. We further demonstrate that optimization of a linear combination of energy and variance can be more efficient than optimizing energy or variance alone.

For demonstration, we use the hydrogen molecule $H_{2}$ as an example. We would consider a more complicated system, $H_{4}$, in Sec. IV. All Hamiltonians are calculated with OpenFermion McClean et al. (2020), and the numeral simulation for variance VQE is conducted with the HiQ simulator framework HiQ . Under STO-3G basis, we use four qubits to describe the Hamiltonian of $H_{2}$. An UCC ansatz is taken with trotter step $k=1$ (see Appendix. A), and there are $9$ parameters (6 for single excitations and 3 for double excitations). The reference state is chosen as $|0011\rangle$, or others in the subspace of two electrons.

By now, we have demonstrated that variance-VQE can efficiently solve excited states for a Hamiltonian. However, the overload of calculating variance and its gradient descent is massive since it scales with the number of Hamiltonian terms as $O(N^{2})$. In this section, we propose stochastic gradient descent for variance-VQE by Hamiltonian sampling, which can significantly reduce the overload.

The Hamiltonian sampling randomly chose some components of $\boldsymbol{c}$ (set $c_{i}=0$ if $c_{i}$ if not chosen), which is denoted as $\tilde{\boldsymbol{c}}$. Then, the estimated energy variance is

A prefactor $\frac{|\boldsymbol{c}|^{2}}{\tilde{\boldsymbol{c}}^{2}}$ is added to account for the fact that coefficients for different terms of a Hamiltonian varies largely. Whenever $\tilde{c}_{i}=0$, elements $\mathcal{G}(\boldsymbol{\theta})_{ij}$ do not need to be evaluated. For a sampling rate $s$ (defined as a ratio that $c_{i}$ is sampled), the number of elements in $\mathcal{G}(\boldsymbol{\theta})_{ij}$ required to be calculated can be reduced by a factor $s^{2}$.

With a Hamiltonian sampling, the gradient can be estimated as $\frac{\partial\tilde{\varDelta}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}}$, which shall have a distribution due to sampling. With this gradient, a stochastic gradient descent algorithm can be applied for optimizing variance-VQE, which updates parameters as

We apply the stochastic gradient descent for solving eigenstates of $H_{2}$ and $H_{4}$ with variance-VQE. The molecule $H_{4}$ is investigated in a trapezoidal structure. Under sto-6g basis, we use a Hamiltonian of $6$ qubits to describe $H_{4}$. The UCC ansatz is chosen as $k=1$ with $21$ parameters, including $15$ for single-particle excitations and $6$ for double-particle excitations (we only consider double-particle excitations from occupied orbitals to empty orbitals). Note that there are more than one hundred terms for the Hamiltonian, which means that Hamiltonian sampling is badly demanded to evaluate the variance.

Numeral simulation results are displayed in Fig. 5 for solving two eigenstates with orthogonal ansatz. For $H_{2}$, it shows that Hamiltonian sampling with a small sampling rate $s=0.1,0.2$ can have comparable convergent behavior with the case without sampling. However, the variance ceases to converge at small values, due to the fluctuation of energy for a subsystem even when the whole system has zero energy variance. To solve this issue, we can turn to no sampling (sampling rate $s=1$) at the late stage of optimization. For $H_{4}$, similar phenomena can be observed. It is noted that only one ansatz reaches the true eigenstate, while the other fail (a local minimum or the ansatz lacks the capacity to express this eigenstate), which can be seen from the nonzero energy variance. This again verifies that variance-VQE can self-verify whether an eigenstate is obtained.

We can explain why Hamiltonian sampling fails to work around the optimized point of parameters, where gradients by Hamiltonian sampling can give meaningless direction for optimization. This is because energy variance is not additive: if $H=H_{A}+H_{B}$ and the wavefunction is $|\psi\rangle$, then the variance of $H$ is not a summation of variance of $H_{A}$ and variance of $H_{B}$. For example, if $|\psi\rangle$ is an eigenstate of $H$, $|\psi\rangle$ will not be an eigenstate of $H_{A}$ or $H_{B}$, if $A$ and $B$ are entangled. As a result, even at the optimized parameter of zero variance, gradients by Hamiltonian sampling can be finite. Such a property is in sharp contrast to the case of machine learning. Machine learning usually uses a set of independent and identically distributed samples, and the cost function is a summation of independent contributions from all samples. Thus, gradients can be estimated with a batch of samples, giving stochastic gradient descents with well convergence behavior Goodfellow et al. (2016).

From the above discussion, a practical strategy is to use Hamiltonian sampling with a small sampling rate to optimize the variance at a small value, then turn to a large sampling rate or no sampling. Also, one may then use energy-VQE. We note that stochastic gradient descent with few shots of measurements can further reduce the overload Sweke et al. (2019); Kübler et al. (2020), which leaves for further investigation.

In summary, we have proposed the variance-VQE that optimizes the energy variance to obtain eigenstates for a Hamiltonian. Compared with VQE that minimizes the energy which is prone to find ground state, the variance-VQE naturally is suitable for solving and self-verifying arbitrary eigenstates. We have adopted different strategies to solve excited states with the variance-VQE, using molecules $H_{2}$ and $H_{4}$ as examples. Remarkably, it has been shown that optimizing a set of orthogonal ansatzs with the same parameterized circuit can be very efficient for calculating a set of eigenstates. Also, we have demonstrated that minimizing a linear combination of energy and variance can outperform the case of optimizing either energy or variance alone. Moreover, we have proposed stochastic gradient descent for minimizing the variance by Hamiltonian sampling, using only a few terms of the Hamiltonian to estimate the variance and its gradient. It has been shown that the variance can decrease quickly under a small sampling rate. While numeral simulations suggested that the variance fails to converge, Hamiltonian sampling can be useful at the early stage to locate the zone of optimized parameters. Our work has demonstrated that variance-VQE can be useful and practical for variational solving excited states for a quantum system on a quantum computer.