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arXiv:2307.06354 · npj Quantum Information

Faster-than-Clifford simulations of entanglement purification circuits and their full-stack optimization

Vaishnavi L. Addala, Shu Ge, Stefan Krastanov

Published Jan 23, 2025• 6 citations
arXivHTMLPDF

Abstract

Generating quantum entanglement is plagued by decoherence. Distillation and error-correction are employed against such noise, but designing a good distillation circuit, especially on today’s imperfect hardware, is challenging. We develop a simulation algorithm for distillation circuits with per-gate complexity of $${\mathcal{O}}(1)$$ , drastically faster than $${\mathcal{O}}(N)$$ Clifford simulators or $${\mathcal{O}}({2}^{N})$$ wavefunction simulators over N qubits. This simulator made it possible to optimize distillation circuits much larger than previously feasible. We design distillation circuits from n raw Bell pairs to k purified pairs and study the use of these circuits in the teleportation of logical qubits. The resulting purification circuits are the best-known for finite-size noisy hardware and can be fine-tuned for specific error-models. Furthermore, we design purification circuits that shape the correlations of errors in the purified pairs such that the performance of potential error-correcting codes is greatly improved.

DOI
10.1038/s41534-024-00948-0

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Generating quantum entanglement is plagued by decoherence. Distillation and error-correction are employed against such noise, but designing a good distillation circuit, especially on today’s imperfect hardware, is challenging. We develop a simulation algorithm for distillation circuits with per-gate complexity of O(1){\mathcal{O}}(1)O(1) , drastically faster than O(N){\mathcal{O}}(N)O(N) Clifford simulators or O(2N){\mathcal{O}}({2}^{N})O(2N) wavefunction simulators over N qubits. This simulator made it possible to optimize distillation circuits much larger than previously feasible. We design distillation circuits from n raw Bell pairs to k purified pairs and study the use of these circuits in the teleportation of logical qubits. The resulting purification circuits are the best-known for finite-size noisy hardware and can be fine-tuned for specific error-models. Furthermore, we design purification circuits that shape the correlations of errors in the purified pairs such that the performance of potential error-correcting codes is greatly improved.
Graph states form a broad class of multipartite entangled states underlying measurement-based quantum computation, quantum networks, and stabilizer codes. However, systematic entanglement distillation for arbitrary graph states remains challenging because the circuit design space grows rapidly with the number of parties. We introduce a group of Clifford operations that we call "factorized graph-preserving". It enables us to efficiently enumerate and optimize graph-state purification circuits at finite size for realistic noisy hardware. These operations map products of graph-basis states to products of graph-basis states, so their action can be represented as permutations of graph-basis labels. Moreover, this useful gate set admits a compact factorized description determined by simple graph-theoretic features. This structure also allows, after some initial cached precomputation, drastically lower computational complexity for simulating a gate. We further organize these operations over local-complementation (LC) orbits using minimum-edge representatives (MERs), which let us design purification circuits that apply to all locally equivalent graph states (up to a basis change). Using this framework, we optimize noisy finite-size multipartite distillation circuits for several graph-state families. Numerical results show that the resulting graph-preserving circuits can outperform standard recurrence-based purification protocols under realistic gate and measurement noise. Our results establish LC-orbit structure and factorized graph-preserving operations as practical tools for scalable, topology-aware and hardware-constrained graph-state distillation protocol design. Our work can also be interpreted as a graph-based heuristic for finding transversal gates.
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We investigate novel protocols for entanglement purification of qubit Bell pairs. Employing genetic algorithms for the design of the purification circuit, we obtain shorter circuits achieving higher success rates and better final fidelities than what is currently available in the literature. We provide a software tool for analytical and numerical study of the generated purification circuits, under customizable error models. These new purification protocols pave the way to practical implementations of modular quantum computers and quantum repeaters. Our approach is particularly attentive to the effects of finite resources and imperfect local operations - phenomena neglected in the usual asymptotic approach to the problem. The choice of the building blocks permitted in the construction of the circuits is based on a thorough enumeration of the local Clifford operations that act as permutations on the basis of Bell states.
Near-term and early fault-tolerant quantum computing architectures are expected to exhibit highly non-uniform error rates. In particular, local operations within a chip can be substantially more reliable than operations connecting different chips or dilution refrigerators. Such inter-module operations can therefore become a dominant bottleneck, even when quantum error correction is applied. Entanglement distillation provides a natural way to trade additional operations and qubits for higher-fidelity entanglement. Standard distillation protocols, however, are usually formulated in an LOCC resource model, in which several noisy Bell pairs are generated initially and all subsequent processing consists only of local operations and classical communication. Here, we consider a generalized model tailored to modular quantum computing hardware, in which the two modules have access to high-fidelity local operations and to repeated uses of the same noisy inter-module entangling operation during the protocol. We develop practical small-scale entanglement distillation protocols designed to minimize both space and time overhead. Remarkably, our main protocol requires only two qubits per module, yet achieves quadratic error suppression of inter-module errors, assuming local operations are much cleaner. Compared with existing small-scale protocols, our space-optimal protocol provides more space- and time-efficient quadratic error suppression and achieves the best performance in our simulations and experiments on noisy links of current superconducting quantum processors. These results suggest that inter-module-gate-assisted entanglement distillation can be a practical primitive for overcoming noisy links in modular quantum computing architectures.
Noisy hardware forms one of the main hurdles to the realization of a near-term quantum internet. Distillation protocols allows one to overcome this noise at the cost of an increased overhead. We consider here an experimentally relevant class of distillation protocols, which distillntokend-to-end entangled pairs using bilocal Clifford operations, a single round of communication and a possible final local operation depending on the observed measurement outcomes. In the case of permutationally invariant depolarizing noise on the input states, we find a correspondence between these distillation protocols and graph codes. We leverage this correspondence to find provably optimal distillation protocols in this class for several tasks important for the quantum internet. This correspondence allows us to investigate use cases for so-called non-trivial measurement syndromes. Furthermore, we detail a recipe to construct the circuit used for the distillation protocol given a graph code. We use this to find circuits of short depth and small number of two-qubit gates. Additionally, we develop a black-box circuit optimization algorithm, and find that both approaches yield comparable circuits. Finally, we investigate the teleportation of encoded states and find protocols which jointly improve the rate and fidelities with respect to prior art.
High-fidelity quantum entanglement enables key quantum networking capabilities such as secure communication and distributed quantum computing, but long-distance entanglement distribution is limited by noise and loss. Entanglement distillation protocols address this problem by extracting high-fidelity Bell pairs from multiple noisy ones. The primary objective is minimizing the resource overhead: the number of noisy input pairs needed to distill each high-fidelity output pair. While protocols achieving optimal overhead are known in theory, they often require complex decoding operations that make practical implementation challenging. We circumvent this challenge by introducing protocols that use quantum scrambling-the spreading of quantum information under chaotic dynamics-through random Clifford operations. Based on this scrambling mechanism, our protocol maintains asymptotically constant overhead, independent of the desired output error rate ϵ[over ¯], and can be implemented with shallow quantum circuits of depth O(poly log logϵ[over ¯]^{-1}) and memory O(poly logϵ[over ¯]^{-1}). Our protocol remains effective even with noisy quantum gates. By incorporating error correction, our protocol achieves state-of-the-art performance: starting with pairs of 10% initial infidelity, we require only seven noisy inputs per output pair to distill a single Bell pair with infidelity ϵ[over ¯]=10^{-12}, substantially outperforming existing schemes. We demonstrate the utility of our protocols for quantum repeater networks.