01518nas a2200145 4500008004100000245006600041210006500107260001300172520107900185100001701264700002101281700001701302700001601319856003701335 2023 eng d00aQuantum Lego Expansion Pack: Enumerators from Tensor Networks0 aQuantum Lego Expansion Pack Enumerators from Tensor Networks c8/9/20233 a

We provide the first tensor network method for computing quantum weight enumerator polynomials in the most general form. As a corollary, if a quantum code has a known tensor network construction of its encoding map, our method produces an algorithm that computes its distance. For non-(Pauli)-stabilizer codes, this constitutes the current best algorithm for computing the code distance. For degenerate stabilizer codes, it can provide up to an exponential speed up compared to the current methods. We also introduce a few novel applications of different weight enumerators. In particular, for any code built from the quantum lego method, we use enumerators to construct its (optimal) decoders under any i.i.d. single qubit or qudit error channels and discuss their applications for computing logical error rates. As a proof of principle, we perform exact analyses of the deformed surface codes, the holographic pentagon code, and the 2d Bacon-Shor code under (biased) Pauli noise and limited instances of coherent error at sizes that are inaccessible by brute force.

1 aCao, ChunJun1 aGullans, Michael1 aLackey, Brad1 aWang, Zitao uhttps://arxiv.org/abs/2308.0515201774nas a2200145 4500008004100000245007900041210006900120260001500189300001100204490000600215520133600221100001701557700001701574856003701591 2022 eng d00aQuantum Lego: Building Quantum Error Correction Codes from Tensor Networks0 aQuantum Lego Building Quantum Error Correction Codes from Tensor c05/11/2022 a0203320 v33 a

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple codes or states, generalizing code concatenation. More specifically, we represent the complex code constructions as tensor networks built from the tensors of simple codes or states in a modular fashion. Using a set of local moves known as operator pushing, one can derive properties of the more complex codes, such as transversal non-Clifford gates, by tracing the flow of operators in the network. The framework endows a network geometry to any code it builds and is valid for constructing stabilizer codes as well as non-stabilizer codes over qubits and qudits. For a contractible tensor network, the sequence of contractions also constructs a decoding/encoding circuit. To highlight the framework's range of capabilities and to provide a tutorial, we lay out some examples where we glue together simple stabilizer codes to construct non-trivial codes. These examples include the toric code and its variants, a holographic code with transversal non-Clifford operators, a 3d stabilizer code, and other stabilizer codes with interesting properties. Surprisingly, we find that the surface code is equivalent to the 2d Bacon-Shor code after "dualizing" its tensor network encoding map.

1 aCao, ChunJun1 aLackey, Brad uhttps://arxiv.org/abs/2109.0815801239nas a2200121 4500008004100000245002800041210002800069260001500097520092800112100002301040700001701063856003701080 2021 eng d00aQuantum Lattice Sieving0 aQuantum Lattice Sieving c10/26/20213 a

Lattices are very important objects in the effort to construct cryptographic primitives that are secure against quantum attacks. A central problem in the study of lattices is that of finding the shortest non-zero vector in the lattice. Asymptotically, sieving is the best known technique for solving the shortest vector problem, however, sieving requires memory exponential in the dimension of the lattice. As a consequence, enumeration algorithms are often used in place of sieving due to their linear memory complexity, despite their super-exponential runtime. In this work, we present a heuristic quantum sieving algorithm that has memory complexity polynomial in the size of the length of the sampled vectors at the initial step of the sieve. In other words, unlike most sieving algorithms, the memory complexity of our algorithm does not depend on the number of sampled vectors at the initial step of the sieve.

1 aRodrigues, Nishant1 aLackey, Brad uhttps://arxiv.org/abs/2110.1335202027nas a2200133 4500008004100000245005400041210005400095520164000149100002001789700001701809700001401826700001601840856003701856 2018 eng d00aQuantum adiabatic optimization without heuristics0 aQuantum adiabatic optimization without heuristics3 a

Quantum adiabatic optimization (QAO) is performed using a time-dependent Hamiltonian H(s) with spectral gap γ(s). Assuming the existence of an oracle Γ such that γ(s)=Θ(Γ(s)), we provide an algorithm that reliably performs QAO in time Oγ−1minlog(γ−1min) with Olog(γ−1min) oracle queries, where γmin=minsγ(s). Our strategy is not heuristic and does not require guessing time parameters or annealing paths. Rather, our algorithm naturally produces an annealing path such that dH/ds≈γ(s) and chooses its own runtime T to be as close as possible to optimal while promising convergence to the ground state. We then demonstrate the feasibility of this approach in practice by explicitly constructing a gap oracle Γ for the problem of finding a vertex m=argminuW(u) of the cost function W:V⟶[0,1], restricting ourselves to computational basis measurements and driving Hamiltonian H(0)=I−V−1∑u,v∈V|u⟩⟨v|, with V=|V|. Requiring only that W have a constant lower bound on its spectral gap and upper bound κ on its spectral ratio, our QAO algorithm returns m using Γ with probability (1−ε)(1−e−1/ε) in time O˜(ε−1[V−−√+(κ−1)2/3V2/3]). This achieves a quantum advantage for all κ, and when κ≈1, recovers Grover scaling up to logarithmic factors. We implement the algorithm as a subroutine in an optimization procedure that produces m with exponentially small failure probability and expected runtime O˜(ε−1[V−−√+(κ−1)2/3V2/3]), even when κ is not known beforehand.

1 aJarret, Michael1 aLackey, Brad1 aLiu, Aike1 aWan, Kianna uhttps://arxiv.org/abs/1810.04686