Optimal Clifford Initial States for
Ising Hamiltonians

Quantum computing (QC) represents a groundbreaking computational paradigm for solving specific problems that are traditionally impractical to tackle classically. It is anticipated that quantum computers (QCs) will wield significant advantages in areas of critical impact such as cryptography [2], chemistry [3], optimization [4], and machine learning [5].

During the ongoing Noisy Intermediate-Scale Quantum (NISQ) era, we are set to operate with quantum machines equipped with hundreds to thousands of imperfect qubits [6]. In this era, these machines will grapple with limited connectivity and relatively short qubit lifetimes due to design constraints. Noise remains a major obstacle, preventing current quantum computers from outperforming classical computers in nearly all applications. Machines in the NISQ era will be incapable of executing extensive quantum algorithms like Shor Factoring [2] and Grover Search [7]. These algorithms necessitate error correction involving millions of qubits to establish fault-tolerant quantum systems [8]. However, a range of error mitigation approaches [9, 10, 11, 12, 13, 14, 15, 16] have been proposed, enhancing execution fidelity on today’s quantum devices. Nonetheless, the achieved fidelity still falls short of the requirements for most practical applications.

In recent times, there has been a growing focus on leveraging classical computing support to elevate the applicability of NISQ applications and devices in the real world. This effort encompasses various enhancements, such as optimizations at the compiler level [17], advancements in classical optimizers [18], circuit segmentation with classical compensation [19, 20], among others. We are currently in the early stages of exploring this synergistic quantum-classical paradigm. There exists significant potential for employing sophisticated classical bootstrapping tailored to specific applications, thereby pushing the boundaries of NISQ capabilities forward.

Variational quantum algorithms (VQAs) are anticipated to align well with NISQ machines, demonstrating a broad array of applications, including electronic energy estimation for molecules [21] and approximations for MAXCUT [4]. The quantum circuit in a VQA is defined by a set of angles, which are fine-tuned by a classical optimizer across multiple iterations to reach a specific target objective representing the VQA problem. These algorithms exhibit greater suitability for present-day quantum devices due to their ability to adapt to the idiosyncrasies and noise profile of the quantum machine [21, 22]. Regrettably, the accuracy achieved by VQAs on existing NISQ machines, even with error mitigation strategies, frequently falls significantly short of the exacting accuracy demands, particularly in domains like molecular chemistry, especially when dealing with larger problem sizes [13, 3, 17].

To advance NISQ VQAs towards practical utility, it is crucial to carefully select the parameterized circuit (ansatz) for a VQA and optimize its initial parameters classically to be as close to optimal as possible before venturing into quantum exploration. This approach holds the promise of enhancing accuracy and expediting algorithmic convergence on the quantum device, even in the presence of noise [23, 24]. While certain applications may derive advantages from domain-specific knowledge guiding the choice of particular parameterized circuits and initial parameters (e.g., UCCSD [25]), these choices are less appropriate for execution on contemporary quantum devices due to their substantial quantum circuit depth. Ansatz circuits tailored for today’s devices, often termed as “hardware efficient ansatz” [3], are typically agnostic to specific applications and stand to benefit significantly from judiciously chosen initial parameters. However, accurately estimating these parameters through classical means can be a challenging task.

Prominent prior work like CAFQA [1] focuses on initializing the VQA ansatz through classical simulation. In CAFQA, the initial parameters for VQA are selected through an efficient and scalable search within a classically simulable segment of the quantum space known as the Clifford space, employing Bayesian Optimization. CAFQA attains remarkable accuracy during initialization. Specifically, for the crucial chemistry application involving the estimation of the ground state energy of molecules, CAFQA restores up to 99.99% of the accuracy lost in previous state-of-the-art classical initialization, demonstrating mean improvements of 56x.

Motivated by this result, in this paper we seek to analyze the Clifford states that optimize the cost function for a new type of Hamiltonian, namely Transverse Field Ising Hamiltonians. Our primary result is contained in theorem IV.1 which connects the problem of finding the optimal Clifford initialization to a submodular minimization problem which in turn can be solved in polynomial time. This connection arises by mapping the problem of finding the Hamiltonian ground state to a graph theoretic optimization problem, which is submodular. Submodular functions satisfy a criterion formalized in Appendix B-C. As a result of this property, submodular functions can be minimized by considering related constrained convex optimization problems [26]. When running numerical experiments, we consistently obtain Clifford approximations of the ground state energies that have a relative error varying from roughly 0-25% when compared to the exact ground state energies. This error itself is problem dependent and is an indicator of the true ‘quantum-ness’ of the problem. Once the good Clifford initialization is found, VQE can be run on future quantum devices (with reasonably low error rates), to find (nearly) exact ground state energy estimates.

Here, we provide a brief overview of VQE. For a more in-depth overview of VQE see [27]. The variational principle from quantum mechanics is a statement regarding the expectation value of a Hamiltonian. Given a Hamiltonian $H$ with ground state $E_{0}$, any quantum state $|\psi\rangle$ will satisfy

Thus if the state can be parameterized by a vector parameter angles $\theta\in[0,2\pi)^{k}$, then

Assuming that the parameterization of $|\psi\rangle$ is expressive enough to accurately predict the ground state of the Hamiltonian.

A common way to parameterize $|\psi\rangle$ is by using a parameterized quantum circuit which implements some parameterized unitary operator $U(\theta)$ acting on the initial state $|0\rangle^{\otimes n}$ where $n$ is the number of qubits.¹¹1From here on we use the shorthand $|0\rangle^{\otimes n}=|0\rangle$. Generally, context will make it clear how many qubits we are considering. Using this parameterization,

and

The circuit which implements the unitary operator $U$ is called the ansatz, and the choice of ansatz is critial to the performance of VQAs [28].

The Variational Quantum Eigensolver (VQE) begins with some set of parameters $\theta_{i}$ and some ansatz $U(\theta)$. It then repeatedly computes the expectation value of the Hamiltonian on $U(\theta)|0\rangle$ using a quantum circuit derived from the ansatz. Between each computation, the parameters are updated using a classical optimizer. The benefit of using quantum devices to find expectation values is that the gate depth of such circuits (both the gates required to construct the ansatz and those required to compute the expectation value once the ansatz has been applied) grows polynomially with $n$ whereas using purely classical simulation will require matricies of size $2^{n}\times 2^{n}$ at minimum.

CAFQA’s primary proposal is to make use of an ansatz which lends itself to a classical initial search before requiring the usage of a quantum computer [1]. It has been shown that circuits which are composed entirely of Clifford Gates can be simulated in polynomial time using the stabilizer technique [29][30]. The states obtained from applying Clifford Gates are called Stabilizer states or Clifford states.²²2We will use these two terms interchangeably.

From here on, we will assume that the ansatz under consideration will be able to reach every $n$-qubit clifford state. With this assumption, determining the best CAFQA initial value is the same as determining the best clifford state initial value.

From here on

• •

  $|\psi\rangle$ will represent an arbitrary state

• •

  $|\varphi\rangle$ will represent an arbitrary Clifford state.

• •

  $|\psi_{0}\rangle$ will be the ground state for a given Hamiltonian $H$

• •

  $|\varphi_{0}\rangle$ will represent the Clifford state that minimizes $\langle\varphi|H|\varphi\rangle$ over all Clifford states $|\varphi\rangle$.

Before moving onto determining the optimal clifford states for a given Hamiltonian, we require the development of some prerequisite theory.

Given a graph $G$, the Transverse Ising Hamiltonian on graph $G$ can be defined as the following $N$-qubit Hamiltonian,

For our purposes we rescale this Hamiltonian as

for some non-negative dimensionless $g=\frac{h}{J}$. This rescaling has the effect of normalizing all eigenvalues by $J$. An overview of Transverse Ising models on $L_{n}$ and $P_{n}$ can be found in [33] while an overview of Transverse Ising Models on random graphs can be found in [34]. A more general overview can be found in [35].

Minimizing the expectation value of $H$ acting on Clifford state $|\varphi\rangle$ then corresponds to minimizing the following,

Because all the Pauli operators above are Hermitian, every expectation value in the above expression is either $\pm 1$ or $0$. The following theorem defines a method for computing $\langle\varphi_{0}|H|\varphi_{0}\rangle$ by solving a related graph theoretic problem.

This theorem will follow from the proofs of Lemmas IV.2 and IV.3 below.

The benefit of mapping the problem of finding $|\varphi_{0}\rangle$ to minimizing $-|E(S)|-g|V_{G}\backslash S|$ is that the latter is a submodular function of $S$. It’s well known that there exists a polynomial time algorithm that can be used to minimize these functions. Appendix B discusses algorithms which can be used to minimize such functions.

We can also maximize over all subsets of a fixed size, which gives the following corollary relating this result to edge functions,

While computing edge functions for specific values of $n$ is still NP-hard, maximizing functions of the above form is always possible in polynomial time.

Appendix B discusses algorithms connected to minimizing submodular functions and computing edge functions.

There are two regimes of $g$-values for which $H$ can be approximated as Hamiltonians for which the ground state is clearly Clifford.

• •

  If $g\to 0$, then

  $$H\approx H_{0}=-\sum_{\langle q_{i},q_{j}\rangle\in E_{G}}Z_{i}Z_{j}$$

  (19)

  $|0\rangle$ is clearly a ground state of $H_{0}$.

• •

  If $g\to\infty$, then

  $$H\approx H_{\infty}=-g\sum_{q_{i}\in V_{G}}X_{i}$$

  (20)

  $|+\rangle^{\otimes N}$ is clearly a ground state of $H_{\infty}$.

For convenience, let

Since $N-n$ controls the number of $X_{i}$ terms in $G_{\varphi}$, we expect that for small $g$, $n=N$ will be optimal while for large $g$ we expect $n=0$ to be optimal.

The following Lemma verifies this behavior.

This Lemma has the following corollary

We call graphs for which $\frac{\mathcal{E}(N)}{N}\geq\frac{\mathcal{E}(n)}{n}$ for $0<n<N$ two-segmented because of the result in Corollary IV.6. For two segmented graphs, we call $\frac{\mathcal{E}(N)}{N}$ the transition value.

It turns out, computing the maximal value of $\frac{\mathcal{E}(n)}{n}$ alongside the maximal vertex set associated with this maximum is possible in polynomial time as discussed in Appendix B.

In this section we compare the optimal Clifford solution via our proposed method to ground states of Ising Hamiltonians obtained via exact diagonalization.

In this work we outlined an efficient method for finding the optimal Clifford ground state of a given transverse field Ising Hamiltonian on an arbitrary graph. Our numerical estimates suggest that these optimal Clifford ground states are generally fairly accurate (with errors always less than $25\%$), suggesting that CAFQA-inspired approaches to VQE problems are useful initialization techniques for these Hamiltonians.

This work is funded in part by EPiQC, an NSF Expedition in Computing, under award CCF-1730449; in part by STAQ under award NSF Phy-1818914; in part by NSF award 2110860; in part by the US Department of Energy Office of Advanced Scientific Computing Research, Accelerated Research for Quantum Computing Program; and in part by the NSF Quantum Leap Challenge Institute for Hybrid Quantum Architectures and Networks (NSF Award 2016136) and in part based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers. This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.