Unitary errors, such as those arising from fault-tolerant compilation of quantum algorithms, systematically bias observable estimates. For example, the Solovay-Kitaev algorithm can be used to decompose any single-qubit rotation into a sequence of Clifford and non-Clifford (T) gates, which approximates the desired operation up to a small unitary error, resulting in a bias in the observable value (see left figure above). One way to correct this bias is to use a decomposition with high accuracy (i.e., low unitary error), which requires substantial additional resources—particularly, a high non-Clifford gate count.

Here, we present an alternative method to construct unbiased observable estimators without requiring additional non-Clifford gates.Our method relies on a decomposition of the ideal channel as a mixture of channels undergoing unitary errors. The unitary errors in the decomposition are the same as those in the ideal channel. Combined with Pauli twirling, we can construct approximate channel estimators using channels that undergo arbitrary unitary errors. Using this approach, we construct unbiased observable estimators based on the output of such channels. We then analyze our method and present the conditions under which it can be applied.

Our method requires knowledge of the error channel parameters, which are typically known in the context of fault-tolerant compilation of quantum algorithms. This makes our method a natural fit for such applications. We carry out numerical simulations to demonstrate its utility. Specifically, we use our method to track an observable during the time dynamics of the Ising Hamiltonian. The results (see right figure above) confirm that our method enables accurate estimation of observables for sufficiently large circuits, albeit at the cost of additional measurements. Furthermore, our method yields accurate estimates of expectation values even in cases where physical gates undergo unitary errors due to miscalibration, provided that partial knowledge of the error parameters is available.

Our strategy offers a resource-efficient way to mitigate the impact of unitary errors, improving the estimation of observables in both noisy near-term quantum devices and fault-tolerant implementations of quantum algorithms.

For more details, please refer to the paper: https://arxiv.org/abs/2505.11486.

Quantum Error Correction is essential for fault-tolerant (FT) quantum computation. However, a major hurdle in realizing FT quantum computation is the substantial overhead required to encode information into logical qubits. This resource-intensive process makes it challenging to perform reliable quantum computations, especially with current quantum devices. To address this, we need to explore alternative approaches that can enable reliable computation in the near term.

In this article, we present a framework for designing quantum algorithms tailored for reliable execution on early fault-tolerant quantum computers. Our focus is on approximating the ground states of molecules, a critical problem in chemistry and materials science.

We first develop an efficient classical algorithm to construct approximations of these ground states using stabilizer states - special types of quantum states that are computationally efficient to work with. By testing our algorithm on molecules with up to 36 qubits, we found that our approximations outperformed traditional methods, such as the Hartree-Fock approximation, which is widely used in quantum chemistry. Additionally, we introduced circuit constructions for preparing generalized stabilizer states - a class of states that offer improved approximations in certain cases. While the approximations we construct do not yet meet the accuracy required for many practical applications, they can serve as a valuable starting point for more sophisticated quantum techniques, such as quantum phase estimation and variational quantum eigensolvers.

The next step involves using the stabilizer approximations to construct quantum error detection codes. Specifically, we used codeword stabilized codes to design error detection methods that require minimal resources. These codes enable the preparation of both stabilizer and generalized stabilizer states with higher fidelity using post-selection. Using a simple noise model, we perform noisy simulations and verify the error detection properties of the codes we construct for preparing the states. These simulations suggest that our method can be useful in preparing approximate eigenstates for different molecular systems, albeit with an overhead due to increasing discard rates.

Our work represents a first step toward the design of algorithms suitable for reliable implementation on quantum devices of the near future. Future work will involve improving this approach for better approximations and developing fully fault-tolerant protocols for these approximations.

For more details, please refer to the paper (https://arxiv.org/abs/2410.21125) posted on arxiv.