Life in an imperfect world: While fault-tolerant quantum computing is an ultimate goal, it is far off, so for now we need to find ways to mitigate the errors that creep into quantum computations. Quantum error correction is of key interest, both theoretically and practically. The authors advance the field by considering error-corrected quantum simulation of geometrically local fermionic systems, providing a systematic way to construct error-correcting code using methods from lattice gauge theory, with the gauge operators serving as stabilizers. This method solves two major problems (geometry locality and error mitigation) in near-term quantum simulations of lattice fermion problems.
Life in an imperfect world: While fault-tolerant quantum computing is an ultimate goal, it is far off, so for now we need to find ways to mitigate the errors that creep into quantum computations. Quantum error correction is of key interest, both theoretically and practically. The authors advance the field by considering error-corrected quantum simulation of geometrically local fermionic systems, providing a systematic way to construct error-correcting code using methods from lattice gauge theory, with the gauge operators serving as stabilizers. This method solves two major problems (geometry locality and error mitigation) in near-term quantum simulations of lattice fermion problems.
Utilizing the framework of \mathbb{Z}_2 ℤ 2 lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any \lfloor \frac{d-1}{2} \rfloor ⌊ d − 1 2 ⌋ -qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically.
Quantum error correction protocols protect against local errors by storing logical information non-locally. This poses a challenge: how to design efficient logical gates on the non-local ``hidden'' logical information, and how to implement these gates using the local physical operations. We develop a general description of planar Pauli stabilizer codes and protocols for logical operations in terms of point-like particles called Majorana fermions. Information is stored in the pairwise fermion parities of spatially separated Majorana fermions. The description in terms of Majorana fermions captures not only large distance asymptotics, but also all scales down to the lattice constants. We exploit this locality to densely pack logical information in spacetime. The simplest application is to a static case: dense memory. More importantly, we implement fault-tolerant Majorana motion and leverage this primitive to design braiding-based logical gates. This approach reduces space overhead of logical operations resulting in an improved logical error rate given fixed number of physical qubits. We illustrate a practical use of our approach by designing and benchmarking of 2-qubit Clifford gates. We find numerically that our protocol outperforms lattice surgery in this setting for near-term error rates and realistic device constraints. More generally, introduction of compact motion of Majorana fermions as an efficient computational primitive opens a promising new route for the design of low overhead error correction protocols.
The construction of a large-scale fault-tolerant quantum computer is an outstanding scientific and technological goal. It holds the promise to allow us to solve a variety of complex problems such as factoring large numbers, quick database search, and the quantum simulation of many-body quantum systems in fields as diverse as condensed matter, quantum chemistry, and even high-energy physics. Sophisticated theoretical protocols for reliable quantum information processing under imperfect conditions have been de-veloped, when errors affect and corrupt the fragile quantum states during storage and computations. Arguably, the most realistic and promising ap-proach towards practical fault-tolerant quantum computation are topologi-cal quantum error-correcting codes, where quantum information is stored in interacting, topologically ordered 2D or 3D many-body quantum systems. This approach offers the highest known error thresholds, which are already today within reach of the experimental accuracy in state-of-the-art setups. A combination of theoretical and experimental research is needed to store, protect and process fragile quantum information in logical qubits effectively so that they can outperform their constituting physical qubits. Whereas small-scale quantum error correction codes have been implemented, one of the main theoretical challenges remains to develop new and improve existing efficient strategies (so-called decoders) to derive (near-)optimal error cor-rection operations in the presence of experimentally accessible measurement information and realistic noise sources. One main focus of this project is the development and numerical implementation of scalable, efficient decoders to operate topological color codes. Additionally, we study the feasibility of im-plementing quantum error-correcting codes fault-tolerantly in near-term ion traps. To this end, we use realistic modeling of the different noise sources, computer simulations, and most modern quantum information approaches to quantum circuitry and noise suppression techniques.
Gauge theories and quantum error-correcting codes share the same underlying structure: both use constraints to identify a specific subspace of the full Hilbert space. In quantum error correction, these constraints are known as stabilizers, while in gauge theories they correspond to Gauss law. In this work, we consider a family of discrete Abelian lattice gauge theories described by a ZN gauge group with N an arbitrary power of two. In this setting, we find a set of stabilizers for the gauge-invariant subspace which is an alternative to the Gauss operators, and we call them binary Gauss stabilizers. We use this alternative stabilizer group to build practical error-correcting codes exploiting the gauge symmetries of the system without the addition of extra qubits. The applications of our finding are not limited to error correction though. We also provide a new strategy of gauge fixing to remove the redundancies based on our alternative stabilizer, which might provide advantages with respect to already-existing approaches such as the axial gauge. Our results provide new tools to study lattice gauge theories and their quantum simulation, and opens directions for future work at the interface of lattice gauge theory and quantum information.
The most scalable proposed methods of simulating lattice fermions on noisy quantum computers employ encodings that eliminate nonlocal operators using a constant factor more qubits and a nontrivial stabilizer group. In this work, we investigated the most straightforward error mitigation strategy using the stabilizer group, stabilizer postselection, that is very natural to the setting of fermionic quantum simulation. We numerically investigate the performance of the error mitigation strategy on a range of systems containing up to 42 qubits and on a number of fundamental quantum simulation tasks including non-equilibrium dynamics and variational ground state calculations. We find that at reasonable noise rates and system sizes, the fidelity of computations can be increased significantly beyond what can be achieved with the standard Jordan-Wigner transformation at the cost of increasing the number of shots by less than a factor of 10, potentially providing a meaningful boost to near-term quantum simulations. Our simulations are enabled by new classical simulation algorithms that scale with the logical Hilbert space dimension rather than the physical Hilbert space dimension.
Quantum computing stands at the precipice of revolutionizing numerous fields, yet its widespread practical application is obstructed by high error rates, far exceeding those encountered in classical computing. For quantum computers to become viable, these error rates must be significantly reduced, which is a critical challenge in the field. This paper explores the evolving landscape of quantum error correction, focusing on the topological code as a leading strategy for improving the fidelity of quantum computations. topological codes utilize a two-dimensional array of qubits and stabilizer operations to detect and correct quantum errors, offering robust scalability for large-scale quantum systems. The theoretical foundations of topological codes are examined along with their implementation challenges and recent advancements in decoding algorithms. Additionally, stabilizer codes are analyzed and their synergistic role in enhancing quantum error correction schemes. By addressing both theoretical concepts and practical considerations, this study provides insights into overcoming quantum error correction obstacles, moving closer to reliable, fault-tolerant quantum computing systems. The paper also highlights emerging techniques that may shape the future of quantum computing reliability.