The capabilities of present-day NISQ devices are constrained by high noise levels and limited error mitigation of qubits. Hence, optimizing NISQ algorithms to use a limited amount of resources is an essential requirement to reduce the effect of errors. One strategy is to use a hybrid classical-quantum approach that employ shallow quantum circuits to solve the “hard” part of the problem. These shallow circuits are more resilient to noise by construction and require limited resources. Variational Quantum Algorithms (VQA) such as the Variational Quantum Eigensolver(VQE) and Quantum Approximate Optimization Algorithm(QAOA) have shown great promise in exploiting this hybrid approach, where a cost function is evaluated in the quantum circuit whose parameters are optimized by a classical non-linear optimizer. The application areas cover an extensive range including chemistry, machine learning, circuit compilation, and classical optimization among others.
Although VQA algorithms are shown to be resilient against coherent errors, the qubits utilized in the quantum circuit are still affected by decoherence. Also, due to high qubit requirements, quantum error correction methods cannot be utilized with VQAs to overcome the effect of decoherence. Another significant source of error that limits the current capability of quantum devices is the readout error caused by the imperfect measurement devices. To combat the noise, one can have the circuit evolution taking place only on a subspace of the full Hilbert space, which will lead to valid and invalid measurement outcomes. The invalid measurement outcomes are attributed to the noise and are discarded. Another helpful approach in combating noise is encoding the data in a format that indicates errors in a straightforward way. One popular approach is one-hot encoding, which results in one-hot quantum states. Such type of binary encodings are used to obtain qubit-saving formulations for the Travelling Salesman Problem, graph coloring problem, quadratic Knapsack problem, and Max k-Cut problem.
In this paper, the authors propose schemes for error mitigation in variational quantum circuits through mid-circuit post-selection, which is performed by injecting the error mitigation sub-circuit consisting of gates and measurements, that detects errors and discards the erroneous data. The work presents post-selection schemes for various encodings which were used to obtain different valid subspaces of quantum states to be used with VQA while solving particular combinatorial optimization problems and problems from quantum chemistry. The encoding methods are i) k-hot, ii) one-hot iii) domain wall encoding iv) Binary and gray encoding and v) one-hot and binary mixed. The measurement was done via two approaches: post-selection through filtering and post-selection through compression. The advantage of the second approach is that it does not require ancilla qubits.
The work implements one-hot to binary post-selection through a compression scheme to solve the Travelling Salesman Problem (TSP) using the Quantum Alternating Operator Ansatz (QAOA) algorithm. In the case of the one-hot method, encoding works by compressing the valid subspace to a smaller subspace of quantum states and differentiates from the known methods. The experiment results show that for amplitude damping, depolarizing, and bit-flip noise, the mid-circuit post-selection has a positive impact on the outcome compared to just using the final post-selection as a criterion. The proposed error mitigation schemes are qubit efficient i.e. they require only mid-circuit measurements and reset instead of a classical “if” operation. The presented methods are currently applicable to existing NISQ algorithms, as well as outside the scope of VQA and with different objective Hamiltonians. Furthermore, mid-circuit measurements have been increasingly made available by providers of quantum hardware such as IBM and Honeywell.
The ancilla-free post-selection through compression scheme can be applied to any problem where the feasible states are one-hot, including the problems defined over permutations such as Vehicle Routing Problem, variations of TSP, Railway Dispatching Problem, Graph Isomorphism Problem, and Flight Gate Assignment Problem. There are multiple factors that should be considered when designing such schemes, including the complexity of the post-selection, the form of the feasible subspace S, the strength, and form of the noise affecting the computation. It is advantageous to design methods that would choose the optimal number (and perhaps the position) of mid-circuit post-selections to be applied automatically. Utilizing such optimized error mitigation schemes can lead to high level of cancellation of noise in NISQ devices.