Holographic Simulation of Correlated Electrons on a Trapped-Ion Quantum Processor

We develop holographic quantum simulation techniques to prepare correlated electronic ground states in quantum matrix-product-state (QMPS) form, using far fewer qubits than the number of orbitals represented. Our approach starts with a holographic technique to prepare a compressed approximation to electronic mean-field ground states, known as fermionic Gaussian matrix-product states (GMPSs), with a polynomial reduction in qubit and (in select cases gate) resources compared to existing techniques. Correlations are then introduced by augmenting the GMPS circuits in a variational technique, which we denote GMPS+X. We demonstrate this approach on Quantinuum’s System Model H1 trapped-ion quantum processor for one-dimensional (1D) models of correlated metal and Mott-insulating states. Focusing on the 1D Fermi-Hubbard chain as a benchmark, we show that GMPS+X methods faithfully capture the physics of correlated electron states, including Mott insulators and correlated Luttinger liquid metals, using considerably fewer parameters than problem-agnostic variational circuits.

Popular Summary

Quantum computers are expected to make a transformative impact on materials science and chemistry by enabling the accurate simulation of complex material properties, molecules, and chemical reactions that lie beyond the reach of even the most powerful conventional supercomputers. However, existing quantum processors have limited memory and are susceptible to noise, creating a large gap between the long-term potential for quantum computation and the capabilities of existing quantum technology.

In this work, we introduce a method to approximately compute the properties of interacting many-electron models, relevant for many materials and chemistry, using an efficient compressed representation called a quantum tensor network. This method both greatly reduces the quantum memory size required to simulate complex materials and molecules, and it also shortens the computation time (reducing the chance for noise to cause errors). Our approach begins by designing quantum circuits to implement an approximate representation of the ground state that can be efficiently calculated using a classical computer. Then we variationally optimize a quantum circuit to build in quantum correlations on top of this classical approximation. A key advantage of this two-step sequence is that it greatly reduces the number of free parameters that must be optimized in the quantum circuit—a key bottleneck for variational quantum algorithms.

We experimentally implement this technique on Quantinuum’s trapped-ion quantum processor, and we benchmark our approach against exact solutions of the Fermi-Hubbard chain—a paradigmatic model of interacting electron physics and quantum magnetism. The efficiency of our approach enables quantitatively accurate calculation of this model using only half as many qubits as electron orbitals being simulated, highlighting the efficiency of the quantum tensor network methods. We also show, through simulations, how these results can be extended to capture two-dimensional and three-dimensional models and to capture thermal properties and phenomena such as superconductivity.

Figure 1

Compressing Gaussian fermion states as QMPS—(a) holographic QMPS implementation (top) of the GMPS circuit (bottom) for approximately preparing compressed Gaussian fermion states. (b)–(c) Experimental implementation of holographic GMPS algorithm for a spinless two-leg ladder at half-filling. (b) (Real part of) Green’s function, $\mathrm{Re}G[\mathbf{r}=(0,0),{\mathbf{r}}^{\prime }=(x,y)]$ with experimental data (orange dots with $1\sigma$ error bars from $1000$ measurement shots per point), noisy circuit simulations with one-qubit and two-qubit gate depolarizing parameters $p_{1q}={10}^{-3}$ and $p_{2q}=5\times {10}^{-3}$, respectively (dashed orange line), and exact values (solid blue line). (c) Connected part of density-density correlators $C_{nn}(\mathbf{r},{\mathbf{r}}^{\prime })$ with $\mathbf{r}=(x_1,0)$ and ${\mathbf{r}}^{\prime }=(x_2,y)$, with $y=0$ data shown as solid lines (theory) and triangles (experiment), and $y=1$ data shown as dashed lines (theory) and circles (experiment), respectively. Each point represents $5600$ measurement shots. Statistical error bars in (c) are included, but smaller than plot symbols.

Figure 2

Variational circuit architectures for correlated electron problems—the GMPS+X approaches augment the GMPS circuit preparing the Hartree-Fock ground state with non-Gaussian gates that build in correlations either by (a) GMPS+J: introducing an extra QMPS layer with an extra bond qubit (gray dashed box) or by (b) GMPS+U: generalizing the GMPS gates (blue boxes) to include non-Gaussian operations, $Z{Z}_{\theta }=e^{-\frac{i}{2}\theta Z\otimes Z}$ and $Z_{\varphi }=e^{-\frac{i}{2}\varphi Z}$. Here we draw the GMPS+X circuits only for block size $B=3$ (the implemented circuits have twice this block size, $B=6$ to include spin). (c) A problem-agnostic brick QMPS ansatz consisting of a brickwork of general number-conserving two-qubit gates. Here, for any Hermitian operator $O$, $O_{\theta }$ denotes a gate corresponding to unitary $u[O_{\theta }]=e^{-i\theta O/2}$.

Figure 3

Comparison of variational approaches to the Fermi-Hubbard chain—relative error in energy of (a) half-filling (b) (1/3)-filling Fermi-Hubbard chain for various variational approaches compared to the exact ground state of the corresponding Fermi-Hubbard chain. Variational approaches include two types of GMPS+X circuits: $\mathrm{GMPS}+\mathrm{J}$ and $\mathrm{GMPS}+\mathrm{U}$ that augment the GMPS with $B=6$ for the Hartree-Fock ground state with additional variational circuitry, and a problem-agnostic QMPS ansatz with two bond qubits and four layers of brick circuit (brick QMPS) (see Appendix pp2 for details of circuit ansatzes). The GMPS+U simulations are performed for an $L=18$ site chain, whereas the brick QMPS and DMRG calculations are performed for an infinite chain. For comparison to classical methods, we include the mean-field (MF) solution, and DMRG results with bond dimension $2^5$ that provide a lower bound on the achievable energy with $n_b=5$ bond qubits ($n_b=B-1$ in general) used in the GMPS+U approach. The number of variational parameters per (spinful) site are, respectively, 1 (GMPS+J), 30 (GMPS+U), 80 (brick QMPS), and $1024$ (DMRG with $\chi =2^5$).

Figure 4

Scaling of the GMPS+U method when changing block size $B$—variational energy of GMPS+U ansatz for the Fermi-Hubbard chain at half-filling with $U=2$ converges rapidly (approximately exponentially over the range of parameters explored) with block size $B$, which suggests that this method can be used to achieve the arbitrary desired accuracy.

Figure 5

Scaling of the GMPS+U ethod when changing system size $L$—the relative error in energy versus the total length $L$ of spinful sites for the FH model at $U=2,4,6$ and $\nu =\frac{1}{2}$ and fixed block size $B=6$.

Figure 6

GMPS+J hardware implementation—for the Fermi-Hubbard chain at half-filling with $U=4$. (a) Energy density versus position in the GMPS+J ansatz (which is not explicitly translation invariant), with hopping energies shown at half-integer (bond-centered) positions. (b) Connected spin ($S$) and charge density ($n$) correlators show spin-charge separation with rapid decay of charge and slower (antiferromagnetically modulated) decay of spin correlations.

Figure 7

GMPS compression resource scaling: (a)–(d) show the mean error $G_{\mathrm{err}}$ of Green’s functions versus the GMPS compression block size for various systems and system sizes. $G_{\mathrm{err}}$ is defined by $G_{\mathrm{err}}=\sum _{i,j}|G_{i,j}^c-G_{i,j}^o|/V$, where $G^o$ and $G^c$ are the original and compressed Green’s function and $V$ is the system size. In each plot we stop at the block size when the eigenvalues cannot be improved by increasing block size as it already reaches the limit of machine precision ${10}^{-15}$. (a)–(c) show the error versus the block sizes for different $L_x$ while $L_y$ is fixed (to 1,6,6); (d) shows the compression error versus the block sizes for different $L_y$ with $L_x=50$. The insets of (a),(b),(d) show the $G_{\mathrm{err}}$ versus $1/L_x$ or $1/L_y$. The inset of (c) shows the $G_{\mathrm{err}}$ versus the block size rescaled by $\log (L_x)$.

Figure 8

Error mitigation. (a) To mitigate hardware errors, we postselect on results with the correct total particle number. To simultaneously measure total particle number and Green’s function elements $⟨c_i^{†}{c}_j⟩$, we add a basis transformation gate $U_M$ for the corresponding qubits of Green’s functions before measuring them and each qubit is measured in the Pauli-$Z$ basis as in Ref. [20]. In the holographic implementation, the first qubit being measured cannot be reset and reused until both sites $i,j$ are measured. (b) A comparison between measurement results for the QMPS VQE method of $U=6$, half-filled Fermi-Hubbard model with (EM) and without (NoEM) the error mitigation. Details of postselection success rate and shot counts are shown in Table 1.

Figure 9

Schematic representation of applying Eq. (B1) to fermionic operator ${\gamma }_{1,N}$.

Figure 10

Brick QMPS method measurement data of Green’s functions and density correlations of the Fermi-Hubbard model of different $U\mathrm{s}$ and fillings. Absolute value of hopping energies $t⟨\frac{1}{2}(c_{i\uparrow }^{†}{c}_{i+1\uparrow }+c_{i\downarrow }^{†}{c}_{i+1\downarrow })⟩$ are shown at bond-centered coordinates with half-integer positions $x=i-0.5$, where $i$ labels the spinful sites. On-site repulsion density correlations $⟨n_{i\uparrow }{n}_{i\downarrow }⟩$ are shown at site-centered coordinates (integer $i$).