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1. Error Correction for Correlated Quantum Systems

   https://doi.org/https://doi.org/10.1007/978-3-030-63591-6_34  

   Byrd, Mark ; Gonzales, Alvin ; Dilley, Dilley ; Thakre, Purva ( August 2021 , International Conference on Applied Mathematics, Modeling and Computational Science, AMMCS 2019: Recent Developments in Mathematical, Statistical and Computational Sciences)

   Kilgour, D.M. ; Kunze, H. ; Makarov, R. ; Melnik, R. ; Wang, X. (Ed.)

   Modeling open quantum systems is a difficult task for many experiments. A standard method for modeling open system evolution uses an environment that is initially uncorrelated with the system in question, evolves the two unitarily, and then traces over the bath degrees of freedom to find an effective evolution of the system. This model can be insufficient for physical systems that have initial correlations. Specifically, there are evolutions ρS=trE(ρSE)→ρ′S=trE(UρSEU†) which cannot be modeled as ρS=trE(ρSE)→ρ′S=trE(UρS⊗ρEU†). An example of this is ρSE=|Φ+⟩⟨Φ+| and USE=CNOT with control on the environment. Unfortunately, there is no known method of modeling an open quantum system which is completely general. We first present some restrictions on the availability of completely positive (CP) maps via the standard prescription. We then discuss some implications a more general treatment would have for quantum control methods. In particular, we provide a theorem that restricts the reversibility of a map that is not completely positive (NCP). Let Φ be NCP and Φ~ be the corresponding CP map given by taking the absolute value of the coefficients in Φ. The theorem shows that the CP reversibility conditions for Φ~ do not provide reversibility conditions for Φ unless Φ is positive on the domain of the code space. 

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2. Designing the Quantum Channels Induced by Diagonal Gates

   https://doi.org/10.22331/q-2022-09-08-802  

   Hu, Jingzhen ; Liang, Qingzhong ; Calderbank, Robert ( September 2022 , Quantum)

   The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of Z -stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal Z -rotation through π / 2 l , that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on l . The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes. 

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3. Connectivity constrains quantum codes

   https://doi.org/10.22331/q-2022-05-13-711  

   Baspin, Nouédyn ; Krishna, Anirudh ( May 2022 , Quantum)

   Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D -dimensional hyperbolic space. 

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4. Guaranteeing completely positive quantum evolution

   https://doi.org/10.1088/1751-8121/ac2e28  

   Dilley, Daniel ; Gonzales, Alvin ; Byrd, Mark ( November 2021 , Journal of Physics A: Mathematical and Theoretical)

   Abstract In open quantum systems, it is known that if the system and environment are in a product state, the evolution of the system is given by a linear completely positive (CP) Hermitian map. CP maps are a subset of general linear Hermitian maps, which also include non completely positive (NCP) maps. NCP maps can arise in evolutions such as non-Markovian evolution, where the CP divisibility of the map (writing the overall evolution as a composition of CP maps) usually fails. Positive but NCP maps are also useful as entanglement witnesses. In this paper, we focus on transforming an initial NCP map to a CP map through composition with the asymmetric depolarizing map. We use separate asymmetric depolarizing maps acting on the individual subsystems. Previous work have looked at structural physical approximation (SPA), which is a CP approximation of an NCP map using a mixture of the NCP map with a completely depolarizing map. We prove that the composition can always be made CP without completely depolarizing in any direction. It is possible to depolarize less in some directions. We give the general proof by using the Choi matrix and an isomorphism from a maximally entangled two qudit state to a set of qubits. We also give measures that describe the amount of disturbance the depolarization introduces to the original map. Given our measures, we show that asymmetric depolarization has many advantages over SPA in preserving the structure of the original NCP map. Finally, we give some examples. For some measures and examples, completely depolarizing (while not necessary) in some directions can give a better approximation than keeping the depolarizing parameters bounded by the required depolarization if symmetric depolarization is used. 

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5. Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and Deletions

   https://doi.org/10.1109/FOCS52979.2021.00077  

   Blocki, Jeremiah ; Cheng, Kuan ; Grigorescu, Elena ; Li, Xin ; Zheng, Yu ; Zhu, Minshen ( February 2022 , IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS))

   Locally Decodable Codes (LDCs) are error-correcting codes for which individual message symbols can be quickly recovered despite errors in the codeword. LDCs for Hamming errors have been studied extensively in the past few decades, where a major goal is to understand the amount of redundancy that is necessary and sufficient to decode from large amounts of error, with small query complexity. Despite exciting progress, we still don't have satisfactory answers in several important parameter regimes. For example, in the case of 3-query LDCs, the gap between existing constructions and lower bounds is superpolynomial in the message length. In this work we study LDCs for insertion and deletion errors, called Insdel LDCs. Their study was initiated by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), who gave a reduction from Hamming LDCs to Insdel LDCs with a small blowup in the code parameters. On the other hand, the only known lower bounds for Insdel LDCs come from those for Hamming LDCs, thus there is no separation between them. Here we prove new, strong lower bounds for the existence of Insdel LDCs. In particular, we show that 2-query linear Insdel LDCs do not exist, and give an exponential lower bound for the length of all q-query Insdel LDCs with constant q. For q ≥ 3 our bounds are exponential in the existing lower bounds for Hamming LDCs. Furthermore, our exponential lower bounds continue to hold for adaptive decoders, and even in private-key settings where the encoder and decoder share secret randomness. This exhibits a strict separation between Hamming LDCs and Insdel LDCs. Our strong lower bounds also hold for the related notion of Insdel LCCs (except in the private-key setting), due to an analogue to the Insdel notions of a reduction from Hamming LCCs to LDCs. Our techniques are based on a delicate design and analysis of hard distributions of insertion and deletion errors, which depart significantly from typical techniques used in analyzing Hamming LDCs. 

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