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arXiv:2110.10280

Logical fermions for fault-tolerant quantum simulation

Andrew J. Landahl, Benjamin Morrison

Published Oct 19, 2021• 4 citations
arXivHTMLPDF

Abstract

We show how to absorb fermionic quantum simulation's expensive fermion-to-qubit mapping overhead into the overhead already incurred by surface-code-based fault-tolerant quantum computing. The key idea is to process information in surface-code twist defects, which behave like logical Majorana fermions. Our approach encodes Dirac fermions, a key data type for simulation applications, directly into logical Majorana fermions rather than atop a logical qubit layer in the architecture. Using quantum simulation of the $N$-fermion 2D Fermi-Hubbard model as an exemplar, we demonstrate two immediate algorithmic improvements. First, by preserving the model's locality at the logical level, we reduce the asymptotic Trotter-Suzuki quantum circuit depth from $\mathcal{O}(\sqrt{N})$ in a typical Jordan-Wigner encoding to $\mathcal{O}(1)$ in our encoding. Second, by exploiting optimizations manifest for logical fermions but less obvious for logical qubits, we reduce the $T$-count of the block-encoding \textsc{select} oracle by 20\% over standard implementations, even when realized by logical qubits and not logical fermions.

DOI
10.48550/arxiv.2110.10280

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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.

quant-pharXiv:2606.03916•Jun 3, 2026•0 citations
We show how to absorb fermionic quantum simulation's expensive fermion-to-qubit mapping overhead into the overhead already incurred by surface-code-based fault-tolerant quantum computing. The key idea is to process information in surface-code twist defects, which behave like logical Majorana fermions. Our approach encodes Dirac fermions, a key data type for simulation applications, directly into logical Majorana fermions rather than atop a logical qubit layer in the architecture. Using quantum simulation of the NNN-fermion 2D Fermi-Hubbard model as an exemplar, we demonstrate two immediate algorithmic improvements. First, by preserving the model's locality at the logical level, we reduce the asymptotic Trotter-Suzuki quantum circuit depth from O(N)\mathcal{O}(\sqrt{N})O(N​) in a typical Jordan-Wigner encoding to O(1)\mathcal{O}(1)O(1) in our encoding. Second, by exploiting optimizations manifest for logical fermions but less obvious for logical qubits, we reduce the TTT-count of the block-encoding \textsc{select} oracle by 20% over standard implementations, even when realized by logical qubits and not logical fermions.
Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC "twist defects" – anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and "logical MZMs," with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.
Quantum error-detecting codes offer a near-term path for improving the performance of quantum simulations on noisy hardware. Using IBM's superconducting quantum computer ibm_boston, we show that partially fault-tolerant encoded quantum simulations of the Ising model in 1+1D and 2+1D outperform their unencoded counterparts in estimating local observables. To represent 42 logical qubits on the heavy-hex quantum processor, 21 blocks of the [[4, 2, 2]] Iceberg code and up to 136 physical qubits are used. By pairing fault-tolerant syndrome extraction with non-fault-tolerant logical operations, this scheme preserves many of the benefits of error detection while avoiding the overhead typically required for a fully fault-tolerant logical gate set. The encoding's square logical connectivity, together with the freedom to place logical qubits within each block, enables simulations of a 2D spatial lattice with lower circuit depth than the unencoded implementation requires. We introduce Observable-Ranked Postselection, a selective-filtering technique based on syndrome correlations that recovers reliable results without the prohibitive shot loss of full syndrome postselection. Under the cumulative effect of device errors, this encoding improves local-observable accuracy over the unencoded baseline by 2-6% at intermediate times in 1+1D simulations, growing with circuit depth to over 200% in 2+1D at the latest times studied.
An important approach to fault-tolerant quantum computation is protecting logical information using quantum error correction. Usually, logical information is in the form of logical qubits, which are encoded in physical qubits using quantum error correction codes. Compared with qubit quantum computation, fermionic quantum computation has advantages in quantum simulations of fermionic systems, e.g., molecules. In this paper, we show that fermionic quantum computation can be universal and fault tolerant if we encode logical Majorana fermions in physical Majorana fermions. We take a color code as an example to demonstrate the universal set of fault-tolerant operations on logical Majorana fermions, and we numerically find that the fault-tolerance threshold is about 0.80.8%0.8.
Fermion lattices can be simulated by same-size qubit lattices with O(1)\mathcal{O}(1)O(1) interaction overhead
Local interactions among electrons underlie many complex properties of correlated materials. While the Jordan-Wigner transformation can preserve this locality along one spatial dimension, interactions along the remaining dimensions typically incur substantial overhead. We show how to simulate all geometrically local interactions on an NNN-site two-dimensional fermion lattice with no asymptotic overhead in the number of interactions and no space overhead. The primary overhead of our method is circuit depth, which on a qubit lattice matches that of fermionic swap networks, scaling as O(N)\mathcal{O}(\sqrt{N})O(N​), but reduces to O(log⁡N)\mathcal{O}(\log N)O(logN) on reconfigurable qubit arrays and to O(1)\mathcal{O}(1)O(1) in lattice-surgery-based surface-code architectures. This is enabled by dynamically reorienting the Jordan-Wigner transformation to switch the lattice dimension along which locality is preserved. Furthermore, we study fermion routing, as required for the simulation of non-local interactions. When using qubit lattices, we reach resource scaling that asymptotically matches that of qubit routing, whilst on fully connected qubit devices, a depth scaling arbitrarily close to O(log⁡N)\mathcal{O}(\log N)O(logN) is reached. This allows the fermionic fast Fourier transform to be implemented on qubit lattices with asymptotically optimal resource scaling under these locality constraints. Notably, all of our constructions naturally extend to ddd-dimensional lattices. Beyond scaling improvements, we show explicit examples of our method, including Fermi-Hubbard-model simulations of the square-, Lieb- and kagome lattice and the fermionic fast Fourier transform.
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.