Product spectrum ansatz and the simplicity of thermal states

Calculating the physical properties of quantum thermal states is a difficult problem for classical computers, rendering it intractable for most quantum many-body systems. A quantum computer, by contrast, would make many of these calculations feasible in principle, but it is still nontrivial to prepare a given thermal state or sample from it. It is also not known how to prepare special simple purifications of thermal states known as thermofield doubles, which play an important role in quantum many-body physics and quantum gravity. To address this problem, we propose a variational scheme to prepare approximate thermal states on a quantum computer by applying a series of two-qubit gates to a product mixed state. We apply our method to a nonintegrable region of the mixed field Ising chain and the Sachdev-Ye-Kitaev model. We also demonstrate how our method can be easily extended to large systems governed by local Hamiltonians and the preparation of thermofield double states. By comparing our results with exact solutions, we find that our construction enables the efficient preparation of approximate thermal states on quantum devices. Our results can be interpreted as implying that the details of the many-body energy spectrum are not needed to capture simple thermal observables.

Figure 1

A tensor network depiction of the PSA on a system of eight qubits. ${\rho }_{\mathrm{prod}}$ is the product spectrum state to which $\ell$ layers of two-qubit unitaries are applied. In this diagram, $\ell =3$.

Figure 2

Plots of exact and PSA free energy over the temperature range $T\in [0.1,5.1]$ with $n=12$ spins and $\ell =4$ layers of unitaries applied to the product spectrum. The free energy is plotted in units of $J$, which we have set to 1.

Figure 3

Exact and PSA subsystem entanglement entropies for the SYK model with $N=20, T=0.7$, and $\ell =16$.

Figure 4

Exact and PSA correlation functions $\langle {\chi }_{\alpha }\left(t\right){\chi }_{\alpha }\rangle$ for the SYK model with $N=20, T=0.7$, and $\ell =16$. $\alpha \in \{4,9,17,20\}$ are displayed as representatives of all the correlation functions.

Figure 5

A tensor network depiction of a local calculation performed on a PSA state. This figure indicates that the expectation value of a one-site observable on a PSA state with a circuit of depth $\ell =3$ can be performed on a subsystem of $n^{\prime }=2\ell =6$ qubits.

Figure 6

Plots of $\langle {\sigma }_i^x\rangle$ (left) and $\langle {\sigma }_i^z\rangle$ (right) for a PSA state of the mixed field Ising chain with $n=100, h_x=1.05$, and $h_z=0.5$ at $T=1.0$.

Figure 7

TFD expectation values: plots of $\langle {\sigma }_i^x\otimes {\sigma }_i^x\rangle$ (upper left), ${\langle {\sigma }_i^x\otimes {\sigma }_i^x\rangle }_C$ (upper right), $\langle {\sigma }_i^z\otimes {\sigma }_i^z\rangle$ (lower left), and ${\langle {\sigma }_i^z\otimes {\sigma }_i^z\rangle }_C$ (lower right) in the mixed field Ising TFD with $n=12,h_x=1.05,$ and $h_z=0.5$ at $T=1.0$. Exact results and PSA results at different layer numbers ($\ell$) are displayed. Here we note that the PSA has been generalized to allow the uncorrelated state ${\rho }_{\mathrm{prod}}$ to be diagonal in an arbitrary local basis.

Figure 8

Exact and PSA TFD correlation functions $\langle {\chi }_{\alpha }\left(t\right)\otimes {\chi }_{\alpha }^{*}\rangle$ for the SYK model with $N=20$ Majorana fermions per subsystem, $T=0.7$, and $\ell =16$. $\alpha \in \{4,9,17,20\}$ are displayed as representatives of all correlation functions.

Figure 9

Exact and PSA (using exact unitary) TFD correlation functions $\langle {\chi }_{\alpha }\left(t\right)\otimes {\chi }_{\alpha }^{*}\rangle$ for the SYK model with $N=20$ Majorana fermions per subsystem and $T=0.2$. $\alpha \in \{4,9,17,20\}$ are displayed as representatives of all correlation functions.

Figure 10

Exact, PSA, and ground-state method (GS) TFD correlation functions $\langle {\chi }_{\alpha }\left(t\right)\otimes {\chi }_{\alpha }^{*}\rangle$ for the SYK model with $N=12, T=0.5$, and $\ell =10$. $\alpha \in \{1,4,7,10\}$ are displayed as representatives of all correlation functions.