Fictitious Copy Quantum Error Mitigation

TL;DR

The reported Fictitious Copy Quantum Error Mitigation method is general purpose for the current generation of quantum devices and is applicable to any problem that measures eigenvalues of operators on sharply peaked distributions.

Abstract

Errors are arguably the most pressing challenge impeding practical applications of quantum computers, which has instigated vigorous research on the development of quantum error mitigation (QEM) techniques. Existing QEM methods suppress errors with a varying degree of efficacy but importantly demand significant additional quantum and classical computational resources. In this work, we present Fictitious Copy Quantum Error Mitigation (FCQEM) method which corrects quantum errors without requiring any additional quantum resources and purely relies on using classical postprocessing of a joint probability distribution to correct expectation values. The joint probability distribution can be measured ``fictitiously'' by sampling one copy of noisy quantum circuit twice, or classically squaring probabilities from simply one copy. We show that FCQEM can recover eigenvalues even if exact eigenstates are not prepared. Furthermore, our technique can benefit other noise mitigation techniques with no additional quantum resources, which is demonstrated by combining FCQEM with the Quantum Computed Moments (QCM) method. FCQEM can compensate for noise that is pathological to QCM, and QCM allows for FCQEM to recover the ground state energy with a larger variety of trial states. We show that our technique can find the exact ground state energy of molecular and spin models under simulated noise models as well as experiments on a Rigetti 84-qubit superconducting quantum processor. The reported FCQEM method is general purpose for the current generation of quantum devices and is applicable to any problem that measures eigenvalues of operators on sharply peaked distributions.

Figures (5)

• Figure 1: (a) Flowchart outlining the Fictitious Copy Quantum Error Mitigation (FCQEM) technique, along with circuit-level descriptions depicted in (b-d). An $n$-qubit state is prepared on a quantum processor and measured in Pauli basis as illustrated in (b), however the noise in quantum processor would severely degrade the state fidelity. In (c) we show a circuit diagram relating FCQEM to a truncated Virtual Distillation approach, but requiring no quantum overhead which equivalent to canceling the noise through classical post-processing step in (d). As will be demonstrated in our work for a variety of implementations on a Rigetti quantum processor, FCQEM recovers the trial state from the noisy distribution in the computational basis.
• Figure 2: Comparison of Virtual Distillation and Fictitious Copy Quantum Error Mitigation on HeH$^+$ under a totally depolarising noise channel before measurement over all four qubits. (a) shows exact state preparation before noise channel; (b) shows applying a Y rotation of $0.2\pi$ on the fourth qubit which prepares the incorrect ground state; (c) shows varying the Y-rotation from the exact ground state at $0$ to $\pi$. Exact refers to exact diagonalisation of the Hamiltonian for the ground state energy. There is no truncation error for exact eigenstates, shown here with the ground state. Furthermore, FCQEM flattens the potential energy surface around this state, allowing for inexact states to correct towards the ground state
• Figure 3: Experimental demonstration of FCQEM and QCM on Rigetti Ankaa-3 84-qubit superconducting device for Helium Hydride (HeH$^+$) and 10 site Transverse Field Ising Model (TFIM) Hamiltonians. (a) The qubit layout for Ankaa-3 is given with pink (yellow) colour to indicate qubits used to prepare the trial state for the HeH$^+$ (TFIM) and yellow colour to indicate qubits for TFIM. (b) The UCCS trial state circuit and (c) Néel state circuit. (d) The HeH$^+$ results as the bond distance between He and H atoms increases. FCQEM is able to correct most of the noise but does not recover the exact ground state energies. At short bond lengths, QCM and FCQEM+QCM can recover the energies and, since FCQEM requires no quantum resources, shows they are efficient to combine. At long bond lengths, QCM finds the negative charge state as it is a lower energy state, whereas QCM+FCQEM recovers the correct charge state. Similarly, (e) shows the TFIM results. For no field, FCQEM outperforms QCM as this state is optimal and the Hamiltonian is diagonal, however FCQEM+QCM is required to recover the ground state energy. As the field increases, FCQEM cannot correct past the Néel state, whereas QCM can, and QCM+FCQEM is able to recover the ground state energy for small external fields with worse performance as the Hamiltonian becomes less diagonally dominant.
• Figure 4: Probability distributions from computational measurement of Figure \ref{['fig:noise']}. Distirbutions are shown uncorrected from device and squared with normalisation to implement FCQEM. FCQEM amplifies peaks and suppresses small contributions for each POVM. (a) shows HeH$^+$ with UCCS Ansatz distribution; (b) shows transverse field ising model with Néel state distribution. Blue boxes represent the uncorrected values with orange lines at the FCQEM corrected values. Values below a probability of $0.005$ are omitted for clarity.
• Figure 5: Demonstration of scaling of FCQEM with larger simulations. (a) shows various scales (16-1024 qubits) of the effectiveness of FCQEM at determining a $n$-qubit spin correlation observable $\langle Z^{\otimes n} \rangle$ with respect to the antiferromagnetic Néel state, prepared with noise using a stabiliser simulator. The raw expectation value (dotted line) and FCQEM correction (solid line) are shown as a function of error rate, where we have applied a Pauli noise channel biased toward dephasing (as to mimic NISQ device noise at the $10^{-2}$ level). Each simulation is run with 100 000 shots. (b) shows the 8 qubit H$_2$O molecule Hamiltonian and Ansatz from Jones2024 under varying depolarisation noise. The noise is given as two qubit depolarisation on each CNOT gate with a tenth of the depolarisation on each single qubit rotation. Inset of (b) shows the difference from exact ground state energy on a logarithmic scale. The circuit from Jones2024 does not prepare an exact ground state but QCM was able to recover the ground state energy. Here we show that FCQEM is not only able to find energies with a slightly better accuracy compared to QCM but noticeably the combination of the two methods increases accuracy by two orders of magnitude and is less sensitive to depolarisation noise.