Present and near-term quantum computers could help us to solve important practical problems by preparing increasingly complex quantum states that cannot be simulated classically with realistic levels of resource. However, imperfections in their quantum operations (noise) severely limits their practical applicability.
Two recent works [arXiv:2011.05942, arXiv:2011.07064] have introduced an approach that now enables researchers to exponentially suppress errors in near-term quantum computers without the need to implement prohibitively expensive quantum error correcting codes.
QuESTlink is an elegant tool that allows users to compactly represent and simulate the quantum circuits required for the error suppression scheme. A library of demonstration material can be found in the GitHub repository.
Demonstrating basic concepts [download .pdf, .nb]
This QuESTlink simulation demonstrates how to construct the derangement circuit for two copies of the computational state. First, the two copies in register 1 and register 2 (Reg. 1 and Reg. 2 in the figure below) are entangled by applying a SWAP operation between them that is controlled on an ancilla qubit, the topmost qubit in the figure below. This SWAP operation can be performed via applying 2-qubit SWAP operations between pairs of qubits as illustrated below. We then also apply a controlled version of our observable. In the figure we want to estimate the expectation value of the observable of the product of Pauli operators X1 X2 X3 X4 X5 and thereby we apply controlled-X operations (CNOTs).
This approach can suppress errors quadratically, i.e., by squaring their probabilities. For example, we perform two computations in parallel that would ideally prepare a GHZ state, but the gates involved are noisy. When we estimate the expectation value of our observable using our derangement circuit shown in the figure below, we find that the noise due to the imperfect gates is drastically reduced.
Derangement operators with more than 2 copies [download .pdf, .nb]
One advantage of this technique is that it can be applied generally to any quantum device without having knowledge of the nature of the errors. Another significant advantage is that we can increase its efficacy exponentially by preparing more copies. This notebook demonstrates how we can construct derangement circuits that can be applied to four copies of the quantum state achieving a suppression proportional to the fourth power of the error probabilities. There are many possible constructions, but we can use Mathematica’s extensive features to check the validity of the corresponding permutation patterns.
Compiling the derangement circuit to existing hardware
[download .pdf, .nb]
The controlled-SWAP operation is typically not a native operation in current generations of quantum hardware. We therefore have to find a way to compile it to the least possible number of native hardware operations. This notebook contains an extensive library of recompilations of the derangement circuit into most of the typical hardware-native gatesets. For example, the controlled-SWAP operation can be implemented using only two-qubit controlled-X rotation gates and single-qubit Z rotation gates as
