Schrödinger-Heisenberg Variational Quantum Algorithms

Almost four decades after Richard Feynman put forward the idea of quantum computing [1], the quantum advantage has been experimentally tested recently in the solid state systems[2, 3, 4] and photonic systems [5, 6]. However, those quantum computational advantage works focused on well-defined quantum sampling problems which were not designed practically useful. Therefore, the next important near-term milestone is to find algorithms for noisy intermediate-scale quantum (NISQ) [7] devices to solve non-trivial practical problems that are intractable for classical computation.

One of the most promising NISQ applications is using variational quantum algorithms (VQA) [8, 9] such as the variational quantum eigensolver (VQE) [10] and the variational quantum simulation (VQS) [11] where a quantum circuit is optimized classically to approximate the eigenstate state energy and to simulate the dynamics of a Hamiltonian respectively for tasks that are widely considered in combinatorial optimization problems [12], condensed matter physics [13], and quantum chemistry [14, 15]. A practical advantage of hybrid algorithms is their certain degrees of resilience to noise in the optimization and quantum hardware [8, 16, 17].

Considering the limitations of NISQ devices, VQAs generally use a shallow local unitary circuit (LUC) (Fig. 1(a)) to approximate the target quantum states. States prepared by shallow LUCs however, could be trivial, obeying the entanglement area law [18] which can be well captured by classical tensor networks [19]. Indeed, the Lieb-Robinson bound [20] indicates that the entanglement light cone restricts the propagation of correlations and, therefore, shallow LUC can not generate long-range entanglement. However, the ground states of some Hamiltonians of interest could be highly non-trivial and require a relatively deep LUC with a depth that has linear or even higher scaling with the qubit number [20, 21] such as interacting spins at critical points [22, 23], topological quantum orders [24, 25], and interacting fermions in complex molecules [15]. This is a big challenge for NISQ devices. Indeed, without an effective quantum error correction, the final fidelity of the quantum circuits drops exponentially with the number of gates. For example, a state-of-the-art random quantum circuit with 60 qubits and 24 layers [3] ended up with a cross-entropy benchmarking fidelity as low as $0.037\%$. We thus need to significantly improve the NISQ hardware to implement those VQAs to the desired accuracy.

This situation can be summarized as a tradeoff between the fidelity of the LUC and its expressivity [9] (i.e., the ability for the quantum circuits to “express” a sufficiently large volume of quantum states to include those non-trivial ones). To circumvent this problem, we propose a new framework of VQAs, enhanced by virtual Heisenberg circuits, which can noiselessly increase the effective circuit depth and thus simultaneously improve its expressivity and fidelity. We want to mention that there is a related work by Zhang et al. where their classical neural networks serve for a similar purpose as our virtual Heisenberg circuits [26]. And there is an orbital optimized unitary coupled cluster method [27] that shares a similar idea as ours where they turn single-excitation circuits into a classical processing on chemical Hamiltonians. We call our scheme Schrödinger-Heisenberg (SH) variational quantum algorithms (VQA), which illustrates that the main idea is that, in addition to the physical unitary circuit, $U$, acting on the quantum states in the Schrödinger picture, we bring in a virtual circuit, $T$, acting on the target Hamiltonian $H$ in the Heisenberg picture (see Fig. 1(a)). In the following, we consider SH-VQE as an example, but we note that the algorithm works for general VQAs. In this case, the energy expectation value $E(T,U)=$ $\left\langle 0^{\otimes n}\left|U^{\dagger}T^{\dagger}HTU\right|0^{\otimes n}\right\rangle$ of the system becomes

where the classically calculated transformed Hamiltonian $H_{T}=T^{\dagger}HT$ has the same energy spectrum as $H$. By properly choosing a relatively deep but noiseless $T$, the state $TU\left|0^{\otimes n}\right\rangle$ could explore the Hilbert space far outside the range of $U\left|0^{\otimes n}\right\rangle$ (see Fig. 1(b)) and hence can obtain lower and more accurate ground-state energy than conventional VQE for non-trivial problems. We show a workflow of SH-VQE together with a comparison to conventional VQE in Fig. 1(c). Compared with VQE, both the real Schrödinger circuit $U$ and the virtual Heisenberg circuit $T$ in SH-VQE are parametrized and updated when minimizing the expectation value $E(T,U)$. The key feature of SH-VQE is that only $U$ as a shallow LUC is physically implemented, whereas the relatively deep circuit $T$ is performed virtually and noiselessly using a classical computer.

We first show how to effectively measure $H_{T}$. In general, the target Hamiltonian $H$ could be expressed as a linear sum of multi-qubit Pauli terms $H=\sum_{i=1}^{m}g_{i}P_{i}$, where $P_{i}\in$ $\left\{\sigma_{I},\sigma_{X},\sigma_{Y},\sigma_{Z}\right\}^{\otimes n}$. Then we can measure each $P_{i}$ with a total number of samples $\left(\frac{m}{\epsilon^{2}}\right)\sum_{i}$ $g_{i}^{2}\operatorname{Var}\left[P_{i}\right]$, proportional to the number of terms $m$ in the Hamiltonian [28], to evaluate the energy expectation value within an error of $\epsilon$. Here $\operatorname{Var}\left[P_{i}\right]=\left\langle P_{i}^{2}\right\rangle-\left\langle P_{i}\right\rangle^{2}$. We can similarly measure $H_{T}$, by similarly decomposing each $T^{\dagger}P_{i}T$ into Pauli strings. While most practical Hamiltonians $H$ only contain a polynomial number of terms, this might not be the case for $T^{\dagger}P_{i}T$ or $H_{T}$, after the transformation (See Appendix).

Here we propose a structure of the Heisenberg circuit that also leads to efficiently measurable $T^{\dagger}P_{i}T$ or $H_{T}$. The circuit consists of two parts (Fig. 1(a)), where the first part is an arbitrary Clifford circuit that can be decomposed into a sequence of $O\left(n^{2}\right)$ basic gates from the set $\{\mathrm{H},\mathrm{S},\mathrm{CNOT}\}$, and the second part is a layer of single-qubit gates. The first part realizes discrete gates such as CNOT to build correlations between any two qubits and the second part makes them continuous. The Clifford circuit maps the multi-qubit Pauli group to itself, which conserves the number of terms of the Hamiltonian. Also, the Gottesman-Knill theorem [30] indicates that calculating the transformed Hamiltonian is easy. While the second part might increase the number of terms of the Hamiltonian, the overhead is polynomial for Hamiltonians $H$ consisting of only $k$-weight terms, i.e., the Pauli operators $\left\{\sigma_{X},\sigma_{Y},\sigma_{Z}\right\}$ act on at most $k$ qubits since the weight remains unchanged. We note that one can change this part into other easier or more complex circuits for different Hamiltonians, considering the trade-off between the circuit power and the measurement cost.

We begin to study the expressivity of the circuit in SH-VQE. We consider the expressivity measure using the method of quantum complex projective $t$-design [31], which means that the distribution of the output states has equal moments up to the $t^{\text{th }}$ order to a Haar uniform distributed states from the whole Hilbert space. Intuitively, as illustrated in Fig. 2(a) [32], a higher $t$-design indicates a more uniform and denser state distribution in the Hilbert space, and vice versa. In general, a LUC of depth $O\left(nt^{10}\right)$ is needed to generate a $t$-design [33], and the Clifford circuits can produce a 3-design [34]. Using the tight Page’s theorem [29], we define the logarithmic difference of entanglement entropy as

to identify the order of expressivity of SH-VQE, where $\rho_{n/2}$ is the reduced half system density matrix, $E_{\text{Haar }}$ is the average over Haar random states, and $E_{SH}$ is the average over the quantum states $TU\left|0^{\otimes n}\right\rangle$. If $\Delta_{t}$ increases and approaches 0 , it means that $TU\left|0^{\otimes n}\right\rangle$ is a $t$-design.

Fig. 2(b) shows a comparison of the expressivity of SH-VQE versus VQE through a numerical experiment on a 12-qubit system. In the VQE setting, we run a random LUC at different depths and calculate $\Delta_{t}$ to characterize the $t$-design. In the SH-VQE setting, we implement both the real Schrödinger circuits $U$ and the virtual Heisenberg circuits $T$ which are pure Clifford consisting of 500 random gates from $\{\mathrm{H},\mathrm{S},\mathrm{CNOT}\}$. The key observation for both cases is the critical depths when the $\Delta_{t}$ measure increases to and saturates at around 0. It is evident that the $\Delta_{t}$ curves for SH-VQE rise much more rapidly than that for VQE for all $t$ values from 3 to 12. The rising curve for SH-VQE quickly hits the saturation point at a Schrödinger circuit depth of $\sim 2$, while the VQE curve arrives at a much deep depth of $\sim 36$. This indicates that SH-VQE can effectively reduce the gate depth by more than one order of magnitude to achieve the same level of expressivity. For a higher number of qubits, we expect an even more dramatic advantage, which can be inferred from a qubit-size dependent test of depth reduction as shown in Fig. D.3. The above results indicate that we can use current NISQ hardware to effectively run deep quantum circuits while maintaining high fidelity. Particularly, based on a two-qubit gate fidelity of $99.5\%$, the SH-VQE can allow us to run, for instance, a 12-qubit 4-depth quantum circuit with an output fidelity of $90\%$, which would otherwise demand a two-qubit gate fidelity of $99.95\%$ (currently unrealistic) and depth of 40 in conventional VQE (Fig. D.2). Note that shallow LUCs or Clifford circuits alone can only generate small design orders, but a combination of them can achieve high expressivity.

We consider an example of the XXZ spin model with a periodic boundary condition

to demonstrate a kind of working flow of SH-VQE. At the critical point $\Delta=1$, the $\mathrm{XXZ}$ model is equivalent to the Heisenberg model whose ground state has a logarithmic scaling of entanglement entropy [22, 23], and hence cannot be prepared by a constant-depth LUC. Since we aim to boost the performance of the NISQ experiments, we use the hardware efficient ansatz [14] for the real Schrödinger circuit even though this may lead to barren plateau problems [35], where each circuit layer composes a layer of CZ gates and a layer of parametrized arbitrary single-qubit gates (denoted as $\vec{\theta})$

For the Heisenberg circuit, the single-qubit gate layer is parametrized with parameters $\vec{\phi}$. And we restrict the Clifford part to graph circuits [37] where only commuting $\mathrm{CZ}$ gates are used. We separate the graph circuit into patterns of different connectivity with the same translational invariant (TI) symmetry as $H_{XXZ}$. More concretely, for an $n$-qubit circuit, we can set $\lfloor n/2\rfloor$ elementary graphs (For the $\mathrm{j}^{\text{th }}$ elementary graph, each node $\mathrm{i}$ is connected with node $i+j$. $\lfloor\cdot\rfloor$ is the floor function.). As each elementary graph can be turned on or turned off, the total number of possible patterns is $2^{\lfloor n/2\rfloor}$ and we use a $\lfloor n/2\rfloor$-bit string to label all the possible patterns such as ‘$01001\ldots$’, where 0 means the corresponding elementary graph is turned on whereas 1 means off (Fig. 3a). To efficiently search through an exponentially large space of Clifford gate patterns, we borrow the idea from differentiable quantum architecture search [38], where each elementary TI graph is turned on independently according to a probability described by a two-parameter softmax function [39]. Thus, only $\lfloor n/2\rfloor\times 2$ parameters (denoted as $\vec{\alpha}$ ) are needed to implement the discrete search of the huge Clifford patterns. Therefore, the circuit ansatz for the SH-VQE is

where $\vec{\alpha}$ and $\vec{\phi}$ represent all configurations of the Heisenberg circuit $T$ and $\vec{\theta}$ are the continuous parameters in the single-qubit gates inside the Schrödinger circuit $U$. The parameters $\vec{\alpha}$ are used to generate samples of different circuits and the cost function is the average of the Hamiltonian expectation values of these circuits under the same gate parameters $\vec{\theta}$ and $\vec{\phi}$. The SH-VQE method then optimizes over all the parameters to search for the ground state of the Hamiltonian.

In our numerical simulation, we consider an 8-spin XXZ model with a 4-depth circuit $U$ and 4 elementary TI graphs of the Clifford layer as shown in Fig. 3(a). We show the energy expectation and the evolution of the possibilities of all 16 configurations during the optimization as functions of the number of iterations in Fig. 3(b). When the energy expectation is converged, the probabilities of the candidate circuit structures concentrate on the optimal configuration, the fully connected graph ‘1111’. In Fig. 3(c), we show the optimal energies of all the 16 candidate circuit configurations, which verifies that the optimal configuration is indeed the fully connected graph ‘1111’. We further solve larger models up to 16 spins to show the improvement of SH-VQE compared with conventional VQE using the same Schrödinger circuits (Fig. 3(d)). For SH-VQE, we directly use the generalized fully connected graph circuits as the Clifford part. We can find under the same circuit depth, the SH-VQE obtains higher fidelities than the VQE (an average improvement of $25.2\%$).

To further demonstrate the practical values of our algorithm, we implement our algorithm to solve the electronic structure problems of $\mathrm{H}_{4}$ and $\mathrm{H}_{2}\mathrm{O}$ molecules following the same workflow as the above. The $\mathrm{H}_{4}$ molecule corresponds to an 8-qubit Hamiltonian. For the $\mathrm{H}_{2}\mathrm{O}$ molecule, we use the active space method [40] to create an effective 10-qubit Hamiltonian containing 10 spin orbitals and 6 electrons. Since the SH-VQA has the Pauli weight restriction, we use the Bravyi-Kitaev mapping which transforms an $M$-mode fermionic Hamiltonian to a spin Hamiltonian of $O\left(\log_{2}M\right)$ Pauli weight [41, 40]. Note that the ground states of these molecule Hamiltonians have the correct number of electrons. The results are shown in Fig. 4, where we can see SH-VQE can reach the chemical accuracy ($1.6\times 10^{-3}$) with Schrödinger circuits of much shallower depth than VQE.

We now give some discussions for SH-VQA. First, we want to emphasize that the states $TU\left|0^{\otimes n}\right\rangle$ are both hard to prepare on NISQ devices, as it requires implementing the relatively deep $T$ circuit, and hard to simulate on classical computers, as it can be treated as Clifford circuits with non-stabilizer input states. However, interestingly, within the SH-VQE framework, the operator expectation values under these states can be efficiently evaluated as long as $U$ is classically tractable. Second, we want to talk about the trainability of SH-VQA. A known result is that in general, an ansatz with high expressivity may lead to low trainability [42]. We want to emphasize that the expressivity benchmarked under the very random settings in Fig. 2 should be understood as the achievable expressivity of the NISQ devices enhanced by Heisenberg circuits but not the actual expressivity of the ansatz within the SH-VQA framework for specific problems. Thus, SH-VQA can be understood as a general methodology for improving existing variational algorithms within which biased and trainable ansatzes can be tested. We summarize some strategies in the Appendix.

In summary, we have introduced a novel variational quantum algorithm, the SH-VQA, to efficiently extend the circuit depth of near-term noisy quantum processors. By virtually introducing relatively deep and non-local Clifford circuits, we show that the expressivity of shallow quantum circuits can be significantly enhanced, without sacrificing the fidelity. We use the XXZ model to demonstrate the workflow of SH-VQA and further demonstrate the practical values of SH-VQA by solving small molecules. Our method is directly applicable to current quantum hardware and is compatible with most existing quantum algorithms. Leveraging quantum error mitigation, our work pushes near-term quantum hardware into wide non-trivial applications.

We use the Qulacs [43] and the Qiskit [44] packages for parts of simulations.