Spin-projection for quantum computation: A low-depth approach to strong correlation

Recent advancements on quantum devices have created widespread interest in the development of efficient quantum algorithms. The Variational Quantum Eigensolver (VQE) is one of the most pursued approaches in the NISQ (noisy intermediate-scale quantum) era, where a trial state $|\psi({\bm{\theta}})\rangle$ parameterized by ${\bm{\theta}}$ is variationally optimized to minimize its Hamiltonian expectation value.Peruzzo et al. (2014); O’Malley et al. (2016) Among the several candidates for $|\psi\rangle$, the unitary coupled cluster with singles and doubles (UCCSD)Kutzelnigg (1977); Bartlett et al. (1989); Taube and Bartlett (2006) ansatz has been extensively used as an entangler in the preparation of a trial state from Hartree-Fock (HF) $|\Phi\rangle$, as shown by the following equation:

Here, $t_{ij\cdots}^{ab\cdots}$ and $\hat{\tau}_{ij\cdots}^{ab\cdots}=\left(a^{\dagger}_{a}a^{\dagger}_{b}\cdots a_{j}a_{i}-h.c.\right)$ are real parameters and anti-hermitian pairs of the $k$th excitation and de-excitation operators in a spin-orbital basis. In this work, we use convention for the orbital indices: $i,j$ for the occupied orbitals of $|\Phi_{\rm HF}\rangle$, $a,b$ for the virtual orbitals, and $p,q,r,s$ for the general orbitals. To make the unitary exponential operator programmable on a quantum device, Trotterization is required in practice for non-commutative exponents, i.e., $e^{\hat{\cal T}_{1}+\hat{\cal T}_{2}}\approx\left(\prod_{ai}e^{t_{i}^{a}\hat{\tau}_{i}^{a}/\mu}\prod_{abij}e^{t_{ij}^{ab}\hat{\tau}_{ij}^{ab}/\mu}\right)^{\mu}$.Barkoutsos et al. (2018); Moll et al. (2018); Romero et al. (2019) However, applying UCCSD to strongly correlated systems often triggers large $t$-amplitudes to account for higher excitations, which, in turn, necessitates a large Trotter number $\mu$, thereby requiring a high-depth quantum circuit. Since Trotterization is nothing but an artifact needed to respect the ansatz Eq. (1), many authors have studied the efficacy of a single Trotter step ($\mu=1$).Barkoutsos et al. (2018); Evangelista et al. (2019); Sokolov et al. (2020) More flexible ansätze have been also proposed, where, in the product of exponential operators, the same anti-hermitian excitations may be repeated albeit with different amplitudes.Lee et al. (2019); Grimsley et al. (2019)

To our knowledge, these recent studies on UCC and its variants have vastly neglected the spin properties of the obtained solutions. Thus, their discussions remained somewhat ambiguous. We argue that it is extremely important to monitor $\langle\hat{S}^{2}\rangle$ because, even with a spin-restricted HF (RHF) reference, UCC amplitudes can spontaneously violate spin-symmetry, thereby variationally lowering its energy as opposed to traditional CC. This often happens especially in strongly correlated systems, such as bond dissociations, as will be demonstrated below. Such broken-symmetry (BS) solutions are not physical for non-relativistic Hamiltonians. Moreover, the problems caused by spin-contamination are well known and documented.Handy et al. (1985); Andrews et al. (1991); Tsuchimochi and Scuseria (2010) This so-called symmetry dilemma poses a challenge in quantum computation. As will be shown, except for singles, using spin-free generators cannot fix this problem because each term is not necessarily commutative. Thus, such “spin-adapted” (SA) methods Paldus (1977); Scuseria et al. (1988) still incur Trotterization for large $t_{ij}^{ab}$. Some of the previous studies on the matter have suggested the use of a constrained approach,Andrews et al. (1991); McClean et al. (2016); Yen et al. (2019); Ryabinkin et al. (2019) wherein the Hamiltonian is augmented with a penalty term $\lambda\left(\hat{S}^{2}-s(s+1)\right)^{2}$, where $\lambda\rightarrow\infty$ allows the elimination of spin-contamination. However, whereas $\langle\hat{S}^{2}\rangle$ can be evaluated with the same effort as the energy by measuring $O(n^{4})$ terms in the operator, the measurement of the expectation value of $\hat{S}^{4}$ has a steep $O(n^{8})$ scaling, where $n$ is the number of qubits (i.e., spin orbitals), compared to the $O(n^{4})$ scaling of the bare Hamiltonian. Ryabinkin et al. have attempted to avoid this prohibitive scaling by constraining the average spin $\langle\hat{S}^{2}\rangle$.Ryabinkin et al. (2019) Nevertheless, the constrained approach does not seem particularly suited to spin as the Trotterized UCCSD ansatz is not flexible enough, provoking the symmetry dilemma. The resulting energy can become unreliable depending on the choice of $\lambda$, if the ansatz is not flexible enough to deal with strong correlation.

In this work, we propose an alternative way of preserving the spin of an arbitrary quantum state in a numerically exact, but much less costly, manner, while simultaneously treating strong correlation. To this end, we will let a quantum state break its spin symmetry, which is subsequently recovered by means of symmetry projection. By writing a spin-adapted state as a superposition of rotated broken-symmetry states, the quantum circuit we develop here fully appreciates the advantage of VQE by replacing a potentially long quantum circuit to describe entanglements with many measurements with short circuits.

We organize the present work as follows. In Section II.1, we introduce spin-adapted UCCSD for quantum computing, and in Section II.2, we demonstrate the practical difficulty in the spin-constrained approach to handle the Lagrange multiplier $\lambda$. Section II.3 then introduces the spin-projection technique for VQE as well as the required quantum circuit. Section III provides some illustrative calculations that are challenging for standard UCC. Finally, we draw our conclusions in Section IV.

We have used PySCFSun et al. (2018) to generate molecular integrals, OpenFermionMcClean et al. (2020) to perform Jordan-Wigner transformation of $\hat{H}$ and $\hat{S}^{2}$, and QulacsQul (2018) to construct quantum circuits for simulation on a classical computer. For optimization, the L-BFGS method was used.Virtanen et al. (2020) $N_{g}$ required for the numerically exact projection varies depending on the degree of spin-contamination; we have used $N_{g}=2$ for N₂ and H₂O and $N_{g}=3$ for O, which are found to be sufficient for the $\langle\hat{S}^{2}\rangle$ values to be accurate to at least $10^{-10}$. The exponential operators in Trotterization are applied in the following order: $abij\rightarrow\bar{a}b\bar{i}j\rightarrow\bar{a}\bar{b}\bar{i}\bar{j}\rightarrow ai\rightarrow\bar{a}\bar{i}$, where, in each spin-block, the leftmost virtual label is the outermost loop while the rightmost occupied label is the innermost loop. The Trotter approximation is considered as having been converged with $\mu=10$.

We first assess the improvement that spin-projection has to offer by computing the potential energy curves of the triple bond dissociation of the singlet N₂ ($m=0$). We have used the STO-6G basis set and frozen $1s$ and $2s$ core electrons of atoms, resulting in an active space of $(6e,6o)$. Plotted in Figure 4 are the energy errors from FCI (kcal/mol) for N₂ in a logarithmic scale. The shaded area corresponds to the “chemical accuracy,” i.e., errors less than 1 kcal/mol. In the lower panel, $\langle\hat{S}^{2}\rangle$ of UHF and BS-UCCSD are likewise depicted. All other methods numerically preserve the correct singlet value $\langle\hat{S}^{2}\rangle=0$ and are, therefore, not shown. The energy of BS-UCCSD starts deviating from that of SA-UCCSD by breaking the symmetry restriction in amplitudes. It is found that this so-called Coulson-Fischer point happens at a much longer distance for UCCSD (2.0 Å) compared to that of HF (1.2 Å). This is because strong correlation can be partially captured via exponential excitations in UCCSD, thereby mitigating spin contamination. However, the ansatz only accounts for disconnected higher excitations, e.g., $\hat{\cal T}_{2}^{2}$, resulting in larger errors. On the other hand, it is worth noting that dPUCCD delivers essentially exact results for these systems while guaranteeing the singlet spin; the largest deviation is 0.007 kcal/mol at 1.5 Å. The error is at least one order of magnitude smaller than those of SA-UCCSD and BS-UCCSD.

Both in PHF and PUCCD, higher excitations are treated via spin-projection and orbital-rotations $\hat{K}$, which do not need Trotterization. Hence, it is expected that the doubles amplitudes of PUCCD are small, thereby allowing for a small Trotter error in dPUCCD compared to those of the disentangled UCCSD (dUCCSD). This is demonstrated in Figure 5, where we have plotted the Trotter error $\Delta$ of each disentangled scheme in the dissociation of N₂. In Table 1, we tabulated the norms of singles (either $|{\bf t}|$ or $|{\bm{\kappa}}|$) and doubles amplitudes obtained at $R_{\rm N-N}=2.2$ Å with $\mu=10$ as well as the distances from those of $\mu=1$, as another indicator of the Trotter error. As can be clearly seen, PUCCD gives the minimal $T_{2}$ amplitudes, thereby allowing for an efficient treatment of strong correlation. In contrast, the converged $T_{1}$-amplitudes of BS-UCCSD are considerable and require a re-optimization of amplitudes for different $\mu$. This indicates that BS-dUCCSD is a whole different ansatz from BS-UCCSD, even for a large bond length, $R_{\rm N-N}=2.8$ Å, where both yield almost identical energies, as is evident from the totally different $\langle\hat{S}^{2}\rangle$, i.e., 1.37 for $\mu=1$ and 2.55 for $\mu=10$.

The above test system employs the minimal basis and admittedly neglects most of the dynamical correlation effect. To show that the accuracy of PUCCD is not a fortuitous artifact resulting from the chosen basis, we have also carried out the calculations for the symmetric bond dissociation of H₂O with a fixed angle of 104.5^∘ using 6-31G and STO-6G for oxygen and hydrogen, respectively. Table 2 summarizes the energy errors of each disentangled method from FCI, where the numbers in parentheses imply those obtained with the STO-6G basis for all the atoms for comparison. The presence of non-negligible dynamical correlation leads to substantial errors if strong correlation is not treated appropriately. This is the case for SA- and BS-UCCSD, while PUCCD still remains to achieve the chemical accuracy.

Lastly, we computed the singlet-triplet energy gap of the oxygen atom to show that neglecting spin-symmetry can often result in a catastrophic error. The basis set used is 6-31G with $1s$ frozen core, constituting another example with a balanced mixture of dynamical and strong correlation effects. Table 3 lists the singlet and triplet energies in Hartree, computed with $m=0$ and $2$, respectively, along with their difference $\Delta_{\rm ST}$ in kcal/mol. It is readily clear that both SA- and BS-UCCSD schemes failed to describe strong correlations of the open-shell singlet oxygen. BS-UCCSD with $m=0$ spontaneously breaks spin and ends up in mainly the low-spin triplet state ($\langle\hat{S}^{2}\rangle=1.96$). Spin-projection in dPUCCD enables to capture strong correlation in the singlet oxygen. As a result, its $\Delta_{\rm ST}$ is found to be in excellent agreement with FCI.

In this Letter, we have shown that numerically exact spin-projection can be made feasible within the framework of VQE. It should be emphasized that spin-projection attains remarkable accuracy in exchange for one ancilla qubit, albeit with little overhead in circuit depth (additional $4n$ CNOT gates). We note that one disadvantage of our scheme is the increase in the number of measurements, but it grows only linearly with $N_{g}$ as opposed to standard nonorthogonal methods that show a quadratic scaling.Huggins et al. (2020) Also, the scheme can potentially suffer from errors when a projected state is not suitable for dealing with strong correlation in the system Qiu et al. (2017b); Degroote et al. (2016). Nonetheless, we should once again stress that the spin-projection operator lends itself straightforwardly to any ansatz in order to guarantee the desired spin-symmetry, and therefore is a promising, versatile tool.

Acknowledgements—This work was supported by JSPS KAKENHI grant numbers JP18H03900 and JP20K15231.