Error detection on quantum computers improving the accuracy of chemical calculations

A major milestone of quantum error correction is to achieve the fault-tolerance threshold beyond which quantum computers can be made arbitrarily accurate. This requires extraordinary resources and engineering efforts. We show that, even without achieving full fault tolerance, quantum error detection is already useful on the current generation of quantum hardware. We demonstrate this experimentally by executing an end-to-end chemical calculation for the hydrogen molecule encoded in the [[4, 2, 2]] quantum error-detecting code. The encoded calculation with logical qubits significantly improves the accuracy of the molecular ground-state energy.

Figure 1

Quantum circuits for the preparation of the UCC ansatz and for the measurement of the expectation values in Eq. (3). The two-qubit logical circuit is shown above the corresponding six-qubit encoded circuit. Qubits $a_1$ and $a_2$ are ancillas. (a) The first section of the encoded circuit prepares the $\overline{|00\rangle }$ logical state. Ancilla $a_1$ is used to detect errors during the preparation. (b) The middle circuit sections apply the UCC exponential. We use ancilla $a_2$ to implement the rotation $\overline{R_y^1\left(\theta \right)}$. (c) The last circuit sections measure the expectation values. Gates $R_t$ perform a basis transformation that depends on the measured term.

Figure 2

Variability of results with qubit mapping. (a) Examples of potential-energy curves obtained using two random mappings for each circuit on the 20-qubit Tokyo chip. (b) Chip geometry with highlighted two-qubit mappings 2A and 2B, and six-qubit mappings 6A and 6B.

Figure 3

Measured expectation values of terms in the Hamiltonian for both the two-qubit and the six-qubit circuit. The top panels show the expectation values and the bottom panels show their differences from the exact values. The spread in values of neighboring points is due to shot noise.

Figure 4

Comparison of VQE results obtained with the two-qubit and the six-qubit circuit using the best available qubits. (a) Energy potential curve of the ${\mathrm{H}}_2$ molecule. The exact energy is the lowest eigenvalue of Hamiltonian (1). (b) Difference between the measured energy and the exact energy. The gray band shows the range of chemical accuracy ($1.6\times {10}^{-3}\mathrm{Ha}$). (c) Value of the UCC parameter $\theta$ at the energy minimum.

Figure 5

Energy potential curves of the ${\mathrm{H}}_2$ molecule calculated analytically using the VQE algorithm with the depolarizing noise model. The two-qubit gate error rate is $p=5\%$.

Figure 6

Energy error with the depolarizing noise model for internuclear separation $R=0.75\text{Å}$. The six-qubit encoded circuit performs better for error rates up to about $30\%$.

Figure 7

Comparison of potential-energy curves obtained with raw measurement outcomes and with outcomes corrected for readout errors for both the two-qubit and six-qubit circuits.

Figure 8

Comparison of potential-energy curves obtained with the six-qubit circuit when ignoring the value of ancilla $a_1$ and when postselecting only outcomes with $a_1$ being zero.