Efficient Variational Quantum Simulator Incorporating Active Error Minimization

One of the key applications for quantum computers will be the simulation of other quantum systems that arise in chemistry, materials science, etc., in order to accelerate the process of discovery. It is important to ask the following question: Can this simulation be achieved using near-future quantum processors, of modest size and under imperfect control, or must it await the more distant era of large-scale fault-tolerant quantum computing? Here, we propose a variational method involving closely integrated classical and quantum coprocessors. We presume that all operations in the quantum coprocessor are prone to error. The impact of such errors is minimized by boosting them artificially and then extrapolating to the zero-error case. In comparison to a more conventional optimized Trotterization technique, we find that our protocol is efficient and appears to be fundamentally more robust against error accumulation.

Popular Summary

Quantum computers hold the potential to solve complex problems in chemistry, physics, and materials science that cannot be handled by conventional digital computers. While practical large-scale quantum computing is still a ways off, simple devices could become available in the next few years. Tantalizingly, a mere 50 physical qubits (the quantum equivalent of a digital bit) suffice to enter uncharted territory, where the processor is so complex that modern supercomputers cannot predict its behavior. However, quantum algorithms require error-correcting hardware, which may need millions of qubits. The challenge, then, is to find useful tasks within this gap, involving hundreds or thousands of qubits rather than millions. We propose a novel approach to simulating quantum systems that requires relatively few qubits and yet can tolerate error accumulation.

We follow a hybrid approach by closely coupling a quantum computer with a traditional one. Our proposed algorithm simply tolerates errors rather than needing hardware-level error correction. To do this, it deliberately increases the error rate in order to understand its effect. This allows us to estimate how the system would behave in the absence of errors. Through numerical analysis, we find that the algorithm can perform useful, conventionally unfeasible tasks using a modest number of qubits.

By coupling a classical computer to a quantum processor, our algorithm can substantially reduce the number of qubits needed, which in turn makes it easier to have several quantum processors working in parallel. We believe that this is a promising approach to realizing the first generation of quantum computers.

Figure 1

Hybrid solver of quantum dynamics. (a) Both a quantum computer and a classical computer are used in solving the time evolution of a quantum system. The quantum state is approximated by a trial state $|\mathrm{\Psi }({\lambda }_1,{\lambda }_2,\dots )⟩$. Variational parameters are determined by the classical computer according to the Schrödinger equation. The quantum computer is used to implement a subroutine: Inputs are parameters $\{{\lambda }_k(t)\}$, and outputs are values of certain derivatives required by the main program in the classical computer. (b) Variational parameters are determined iteratively given their initial values. Parameters at the time $t$ are sent to the quantum computer, which evaluates the quantities required by the classical computer. Then, the classical computer can work out parameters at the time $t+\delta t$, where $\delta t$ is a short time. Note that the curves here represent the actual evolution of parameters in the example described in Appendix pp1.

Figure 2

Quantum circuit for the evaluation of certain coefficients required by the classical main program, as specified in the text. To evaluate $\mathrm{Re}(e^{i\theta }⟨\overline{0}|U|\overline{0}⟩)$, where $U=R_1^{†}\dots U_k^{†}{R}_k^{†}\dots R_{N_{\mathrm{v}}}^{†}{R}_{N_{\mathrm{v}}}\dots R_q{U}_q\dots R_1$, the ancillary qubit is initialized in the state $(|0⟩+e^{i\theta }|1⟩)/\sqrt{2}$ and measured in the $|\pm ⟩=(|0⟩\pm |1⟩)/\sqrt{2}$ basis. Here, $U_k$ is one of ${\sigma }_{k,i}$, and $U_q$ is one of ${\sigma }_{q,j}$ or ${\sigma }_j$ (by taking $q=N_{\mathrm{v}}+1$, ${\sigma }_j$ is put on the left side of $R_{N_{\mathrm{v}}}$ in the product). In the figure, we have assumed that $k<q$. Gates on the register after the second controlled unitary gate can be omitted. This circuit is actually a variant of the circuit proposed in 2002 by Ekert et al. [52, 53]. It involves $N_{\mathrm{v}}$ gates on the register, two flip gates ($X$) on the ancillary qubit, and two controlled unitary gates on the ancillary qubit and the register.

Figure 3

The distance between the true wave function and the wave function obtained from the hybrid algorithm. Black circles denote true wave functions given by the exact time evolution at the time $t_0,t_1,\dots ,t_N$, respectively. $U_n$ denotes the exact time evolution during the time from $t_{n-1}$ to $t_n$. Gray circles denote trial wave functions, and each empty circle with a dashed edge is the wave function at the time $t_n$ given by the exact time evolution $U_n$ and taking the trial wave function at the time $t_{n-1}$ as the initial state. Distances are marked by red double lines. Note that $D_a=D(|{\mathrm{\Phi }}_{n-1}⟩,|{\mathrm{\Psi }}_{n-1}⟩)$ and $D_b=D(|{\mathrm{\Phi }}_n⟩,U_n|{\mathrm{\Psi }}_{n-1}⟩)$. The distance between $|{\mathrm{\Phi }}_N⟩$ and ${\rho }_N$ is not larger than the sum of all of the red double lines in the main figure.

Figure 4

(a) A schematic diagram of the error reduction. The true value $⟨X{⟩}^{(0)}$ is inferred by measuring $⟨X⟩$ for a set of error factors $r$ and fitting data using the function $⟨X⟩=⟨X{⟩}^{(0)}+\chi r$. (b) Error twirling and simulation. Nonstochastic errors in a controlled-phase gate can be converted into stochastic errors by performing Pauli gates before and after the gate, and error probabilities can be tuned by applying Pauli gates after the gate.

Figure 5

Numerical estimations of the performance of quantum simulation using the Trotterization algorithm and the hybrid algorithm. The simulated system is a quantum Ising model of three spins initialized in the cluster state. (a) Trace distance $D(|\mathrm{\Phi }⟩,\rho )$ as a function of the simulated time $t$. Here, $|\mathrm{\Phi }(t)⟩$ is the true state, and $\rho (t)$ is the state prepared in the quantum computer. The Trotterization algorithm (black curves) and the hybrid algorithm are compared. In the Trotterization algorithm, $\rho (t)$ is prepared according to the Trotter-Suzuki decomposition, and in the hybrid algorithm, $\rho (t)$ is prepared according to the trial state $|\mathrm{\Psi }(t)⟩$. We have taken the error rate ${\epsilon }^{(2)}=0.1\%$. The hybrid algorithm without the error reduction (blue curves) is already much more reliable than the conventional Trotterization algorithm. In the hybrid algorithm, one can further reduce the distance using the error-reduction protocol (red curve). The residual distance that cannot be eliminated using the error reduction is mainly due to errors in the state preparation (i.e., $D(|{\mathrm{\Psi }}_N⟩,{\rho }_N)$ term). Inset: The parameter $\delta t$ in the Trotterization algorithm is optimized to minimize the average distance, and the gray curve is obtained by using the lowest-order symmetric Trotter-Suzuki decomposition (see Appendix pp4). (b) The difference between the average value of a stabilizer estimated using the quantum computer and its true value. Here, $S(\rho )=\mathrm{Tr}(S_2\rho )$ and $S(|\mathrm{\Phi }⟩)=⟨\mathrm{\Phi }|S_2|\mathrm{\Phi }⟩$. Average values of other stabilizers are only slightly different. The true average value of the stabilizer is plotted in the inset. In both figures (a) and (b), light blue and red bands denote the fluctuation due to shot noise: With 68% chance (in total 100 trials), the corresponding quantity is within the band.