Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits

The vacuum of the lattice Schwinger model is prepared on up to 100 qubits of IBM’s Eagle-processor quantum computers. A new algorithm to prepare the ground state of a gapped translationally invariant system on a quantum computer is presented, which we call “scalable circuits ADAPT-VQE” (SC-ADAPT-VQE). This algorithm uses the exponential decay of correlations between distant regions of the ground state, together with ADAPT-VQE, to construct quantum circuits for state preparation that can be scaled to arbitrarily large systems. These scalable circuits can be determined with use of classical computers, avoiding the challenging task of optimizing parameterized circuits on a quantum computer. SC-ADAPT-VQE is applied to the Schwinger model, and is shown to be systematically improvable, with an accuracy that converges exponentially with circuit depth. Both the structure of the circuits and the deviations of prepared wave functions are found to become independent of the number of spatial sites, $L$. This allows a controlled extrapolation of the circuits, determined with use of small or modest-sized systems, to arbitrarily large $L$. The circuits for the Schwinger model are determined on lattices up to $L=14$ (28 qubits) with the Qiskit classical simulator, and are subsequently scaled up to prepare the $L=50$ (100 qubits) vacuum on IBM’s 127-superconducting-qubit quantum computers ibm_brisbane and ibm_cusco. After introduction of an improved error-mitigation technique, which we call “operator decoherence renormalization”, the chiral condensate and charge-charge correlators obtained from the quantum computers are found to be in good agreement with classical matrix product state simulations.

Popular Summary

Quantum simulations of Standard Model physics are expected to improve our understanding of fundamental aspects of the Universe, from the dynamics a split second after the Big Bang to the jets of hadrons produced in high-energy collisions. While the interactions between the quarks and gluons of the Standard Model are well known, how they lead to the complex, strongly interacting, and correlated quantum systems in nature remains elusive because of the phenomena of confinement and chiral symmetry breaking in quantum chromodynamics (QCD). Fortunately, it is believed that these processes can be probed with Hamiltonian QCD simulations using quantum computers.

This work focuses on the simulation of quantum electrodynamics in one spatial dimension (also known as the Schwinger model), which shares essential features with QCD such as confinement. We develop a new framework, built upon the recently developed quantum algorithm ADAPT-VQE, to prepare low-energy states on the register of a quantum computer and apply it to the preparation of the Schwinger model vacuum. This scalable technique takes advantage of two generic features: invariance under translations in space and the existence of a mass gap. Together, these imply that the state preparation quantum circuits have structure only over a relatively small interval, which is then repeated across the entire qubit register. This insight allows a controlled extrapolation of the vacuum preparation circuits to arbitrarily large systems. These circuits were successfully executed on up to 100 qubits of IBM’s quantum computers, and the measured correlations between electric charges and fermion condensate were found to be in excellent agreement with theoretical expectations.

Figure 1

Pictorial description of the SC-ADAPT-VQE algorithm. Once a pool of scalable operators $\{{\hat{O}}_i\}$ has been identified, ADAPT-VQE is performed with use of classical computers to determine a quantum circuit (parameterized by $\{{\theta }_i\}$) that prepares the vacuum up to a desired tolerance. ADAPT-VQE is repeated for multiple lattice sizes, $\{L_1,L_2,\dots \}$, and the circuit parameters are extrapolated to the desired $L$, which can be arbitrarily large. The extrapolated circuits are executed on a quantum computer to prepare the vacuum on a system of size $L$.

Figure 2

$L$ extrapolations of the vacuum energy density $\epsilon$ (left) and the chiral condensate $\chi$ (right) for $m=0.5$ and $g=0.3$. The results of exact diagonalization calculations for $L\ge 9$ (blue circles) given in Table 1 and DMRG calculations (orange squares) given in Table 5 are extrapolated to $L\to \mathrm{\infty }$, as shown by the darker points. The solid lines correspond to linear extrapolations and the dashed lines correspond to quadratic extrapolations, and are found to overlap (see the insets). The difference between the $L\to \mathrm{\infty }$ values of these two extrapolations defines the (fitting) uncertainties associated with the darker points.

Figure 3

(a) Definition of the $R_{\pm }(\theta )$ gate, which implements $\exp [-i\theta /2(\hat{Y}\hat{X}\pm \hat{X}\hat{Y})]$. The $R_{\pm }(\theta )$ gate is used to implement (b) $\exp [-i\theta /2(\hat{X}{\hat{Z}}^2\hat{Y}-\hat{Y}{\hat{Z}}^2\hat{X})]$ and (c) $\exp [i\theta /2(\hat{X}{\hat{Z}}^4\hat{Y}-\hat{Y}{\hat{Z}}^4\hat{X})]$ (note the change in sign).

Figure 4

Simplifications of quantum circuits for the Trotterized unitaries corresponding to (a) ${\hat{O}}_{mh}^V(1)$, (b) ${\hat{O}}_{mh}^V(3)$, and (c) ${\hat{O}}_{mh}^V(5)$ for $L=6$, as explained in the main text. Cancellations between $R_{+}(\pm \frac{\pi }{2})$ are highlighted with red-dash-outlined boxes.

Figure 5

Deviations from the exact values of the energy density $\delta \epsilon$, chiral condensate $\delta \chi$, and wave function infidelity density ${\mathcal{I}}_L$ obtained with SC-ADAPT-VQE. The deviation in quantity “$x$” is defined as $\delta x=|(x^{(\mathrm{aVQE})}-x^{(\mathrm{exact})})/x^{(\mathrm{exact})}|$, where $x^{(\mathrm{exact})}$ denotes the exactly calculated value at the same $L$. Results are shown for $L=6$ to $L=14$ as a function of step number for $m=0.1$ and $g=0.3$ (left column), $m=0.1$ and $g=0.8$ (center column), and $m=0.5$ and $g=0.3$ (right column). The numerical values for $m=0.5$ and $g=0.3$ for the seventh step (highlighted with the dashed box) are given in Table 1, and the sequencing of the corresponding Trotterized operators and the variational parameters are given in Table 2. The corresponding results for $m=0.1$ and $g=0.3$ (seventh step) and for $m=0.1$ and $g=0.8$ (sixth step) can be found in Appendix pp6.

Figure 6

Example of fitting the asymptotic $L$ dependence of a parameter defining the SC-ADAPT-VQE state-preparation circuit. The results for ${\theta }_1$, corresponding to evolving by ${\hat{O}}_{mh}^V(1)$ (blue circles), determined from classical simulations, for $m=0.5$ and $g=0.3$ given in Table 2, are extrapolated to $L=\mathrm{\infty }$ by (i) use of a three-parameter fit given in Eq. (10), as shown by the blue line, with an asymptotic value shown by the blue region, and by (ii) the forming of effective $\theta$ (orange diamonds) defined in Eq. (D3), with the maximum and minimum values shown as the orange shaded region.

Figure 7

Local chiral condensate ${\chi }_j$ for $L=50$, as obtained from IBM’s Eagle-processor quantum computer ibm_cusco after different steps of error mitigation: DD (squares), PT (diamonds), and ODR (circles). This is compared with the expected results obtained from the Qiskit MPS circuit simulator (black dashes). Averaging ${\chi }_j$ over all of the qubits (including at the boundaries) gives the chiral condensate presented in Table 4. The layout of the qubits used on the processor is shown on the right. These results were obtained by our performing 150 Pauli twirls, each involving $8\times {10}^3$ shots for the physics circuits and the corresponding mitigation circuits. The blue icon in the upper right indicates that this calculation was done on a quantum device [154].

Figure 8

Left: Connected contributions to the spatial charge-charge correlation functions, $⟨{\hat{\overline{Q}}}_j{\hat{\overline{Q}}}_k{⟩}_c$, for $L=50$ (the inset shows the number of standard deviations by which the results obtained from ibm_cusco deviate from the MPS simulator results). Right: Volume-averaged correlation functions as a function of distance $d$, $⟨\hat{\overline{Q}}\hat{\overline{Q}}{⟩}_c(d)$, with the points following the same color map as in the left main panel (error bars show $1\sigma$ standard deviations).

Figure 9

Charge on each spatial site, ${\hat{\overline{Q}}}_k$, for $m=0.5$, $g=0.3$, and $L=14$ obtained from exact diagonalization of the Hamiltonian.

Figure 10

$L$ extrapolations of the vacuum energy density $\epsilon$ (top) and chiral condensate $\chi$ (bottom) for $m=0.1$ and $g=0.3$ (left) and $m=0.1$ and $g=0.8$ (right). Each panel shows extrapolations of the exact values given in Tables 7 and 9 (blue circles) and of the results from DMRG calculations (orange squares) given in Table 5 for $L\ge 9$. The solid lines correspond to linear extrapolations and the dashed lines correspond to quadratic extrapolations, and are found to overlap (see the insets). The darker points show the $L=\mathrm{\infty }$ extrapolated value, with an uncertainty determined by the difference between the linear and quadratic extrapolations.

Figure 11

(a) Orderings of Trotterized terms for implementing ${\hat{O}}_{mh}^V(1)$. Circuits of depth 2–7 are shown from left to right, with the dumbbells representing the circuit in Fig. 3. (b) Deviations in the energy density of the SC-ADAPT-VQE prepared state for $m=0.5,g=0.3$, and $L=10$ with different depth implementations of the Trotterization of ${\hat{O}}_{mh}^V(1)$.

Figure 12

Further examples of fitting the asymptotic $L$ dependence of the variational parameters defining the state preparation quantum circuit, determined from classical simulations using SC-ADAPT-VQE. The results for ${\theta }_1={\hat{O}}_{mh}^V(1)$ (blue points) for $m=0.1$ and $g=0.3$ (left panel) given in Table 6 and for $m=0.1$ and $g=0.8$ (right panel) given in Table 8 are extrapolated to $L=\mathrm{\infty }$ by (i) use of a three-parameter fit given in Eq. (10), as shown by the blue line and shaded region, with an asymptotic value shown by the gray region, and by (ii) the forming of effective $\theta$ (orange diamonds), with the maximum and minimum values shown as the orange shaded region (where possible).

Figure 13

Expectation values of ${\chi }_j$ for $L=14$, 20, 30, and 40 (from top to bottom) obtained from simulations using ibm_brisbane. They are compared with the expected results obtained with use of Qiskit’s MPS circuit simulator (black dashes). Averaging ${\chi }_j$ over all of the qubits provides the chiral condensates presented in Table 4. The layouts of the qubits used on the chip are shown in the insets.

Figure 14

Expectation values of $CP$-averaged ${\chi }_j$ for $L=14$, $20$, $30$, $40$, and $50$ (from top to bottom) obtained from simulations using ibm_brisbane and ibm_cusco. They are compared with the expected results obtained with use of Qiskit’s MPS circuit simulator (black dashes). The layouts of the qubits used on the chip are shown in the insets (with same-colored qubits being averaged).

Figure 15

Connected contribution to the spatial charge-charge correlation functions, $⟨{\hat{\overline{Q}}}_j{\hat{\overline{Q}}}_k{⟩}_c$ (left) and averaged correlation functions as a function of distance $d$, $⟨\hat{\overline{Q}}\hat{\overline{Q}}{⟩}_c(d)$ (right), with the points following the same color map as in the left main panel (error bars show $1\sigma$ standard deviations). Results obtained from ibm_brisbane are shown for $L=14$, $20$, $30$, and $40$ (from top to bottom).

Figure 16

Results for the $L=30$ system obtained with use of three steps of SC-ADAPT-VQE, obtained from simulations using ibm_brisbane with 80 twirled instances. The top panel shows ${\chi }_j$, the middle panel shows the $CP$-averaged ${\chi }_j$, and the bottom panels show the connected contribution to the spatial charge-charge correlation functions, $⟨{\hat{\overline{Q}}}_j{\hat{\overline{Q}}}_k{⟩}_c$ (the first two spatial sites are not shown due to the errors on qubits 0 and 2), and the averaged correlation functions as a function of distance $d$, $⟨\hat{\overline{Q}}\hat{\overline{Q}}{⟩}_c(d)$, with the points following the same color map as in the left main panel (error bars show $1\sigma$ standard deviations).

Figure 17

Results for the $L=50$ system obtained with use of three steps of SC-ADAPT-VQE, obtained from simulations using ibm_cusco with 40 twirled instances. The top panel shows ${\chi }_j$, the middle panel shows the $CP$-averaged ${\chi }_j$, and the bottom panels show the connected contribution to the spatial charge-charge correlation functions, $⟨{\hat{\overline{Q}}}_j{\hat{\overline{Q}}}_k{⟩}_c$ (the first two spatial sites are not shown due to the errors on qubits 0 and 2), and the averaged correlation functions as a function of distance $d$, $⟨\hat{\overline{Q}}\hat{\overline{Q}}{⟩}_c(d)$, with the points following the same color map as in the left main panel (error bars show $1\sigma$ standard deviations).