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In this paper, we study the problem of learning an unknown quantum circuit of a certain structure. If the unknown target is an $n$-qubit Clifford circuit, we devise an efficient algorithm to reconstruct its circuit representation by using $O(n^2)$ queries to it. For decades, it has been unknown how to handle circuits beyond the Clifford group since the stabilizer formalism cannot be applied in this case. Herein, we study quantum circuits of $T$-depth one on the computational basis. We show that the output state of a $T$-depth one circuit {\textit{of full $T$-rank}} can be represented by a stabilizer pseudomixture with a specific algebraic structure. Using Pauli and Bell measurements on copies of the output states, we can generate a hypothesis circuit that is equivalent to the unknown target circuit on computational basis states as input. If the number of $T$ gates of the target is of the order $O({{\log n}})$, our algorithm requires $O(n^2)$ queries to it and produces its equivalent circuit representation on the computational basis in time $O(n^3)$. Using further additional $O(4^{3n})$ classical computations, we can derive an exact description of the target for arbitrary input states. Our results greatly extend the previously known facts that stabilizer states can be efficiently identified based on the stabilizer formalism., Comment: 14 pages, 7 figures. The notion $T$-rank is introduced. Lemma 13 has been modified. Several related statements are revised

Quantum information needs to be protected by quantum error-correcting codes due to imperfect physical devices and operations. One would like to have an efficient and high-performance decoding procedure for the class of quantum stabilizer codes. A potential candidate is Pearl's belief propagation (BP), but its performance suffers from the many short cycles inherent in a quantum stabilizer code, especially highly-degenerate codes. A general impression exists that BP is not effective for topological codes. In this paper, we propose a decoding algorithm for quantum codes based on quaternary BP with additional memory effects (called MBP). This MBP is like a recursive neural network with inhibitions between neurons (edges with negative weights), which enhance the perception capability of a network. Moreover, MBP exploits the degeneracy of a quantum code so that the most probable error or its degenerate errors can be found with high probability. The decoding performance is significantly improved over the conventional BP for various quantum codes, including quantum bicycle, hypergraph-product, surface and toric codes. For MBP on the surface and toric codes over depolarizing errors, we observe error thresholds of 16% and 17.5%, respectively., Comment: 22 pages, 25 figures, 3 tables, and 3 algorithms

A quantum stabilizer code over GF$(q)$ corresponds to a classical additive code over GF$(q^2)$ that is self-orthogonal with respect to a symplectic inner product. We study the decoding of quantum low-density parity-check (LDPC) codes over binary finite fields GF$(q=2^l)$ by the sum-product algorithm, also known as belief propagation (BP). Conventionally, a message in a nonbinary BP for quantum codes over GF$(2^l)$ represents a probability vector over GF$(2^{2l})$, inducing high decoding complexity. In this paper, we explore the property of the symplectic inner product and show that scalar messages suffice for BP decoding of nonbinary quantum codes, rather than vector messages necessary for the conventional BP. Consequently, we propose a BP decoding algorithm for quantum codes over GF$(2^l)$ by passing scalar messages so that it has low computation complexity. The algorithm is specified in log domain by using log-likelihood ratios (LLRs) of the channel statistics to have a low implementation cost. Moreover, techniques such as message normalization or offset can be naturally applied in this algorithm to mitigate the effects of short cycles to improve BP performance. This is important for nonbinary quantum codes since they may have more short cycles compared to binary quantum codes. Several computer simulations are provided to demonstrate these advantages. The scalar-based strategy can also be used to improve the BP decoding of classical linear codes over GF$(2^l)$ with many short cycles., Comment: correct the definition of {\cal S}^\perp on page 4 and equation (17)

Codes based on sparse matrices have good performance and can be efficiently decoded by belief-propagation (BP). Decoding binary stabilizer codes needs a quaternary BP for (additive) codes over GF(4), which has a higher check-node complexity compared to a binary BP for codes over GF(2). Moreover, BP decoding of stabilizer codes suffers a performance loss from the short cycles in the underlying Tanner graph. In this paper, we propose a refined BP algorithm for decoding quantum codes by passing scalar messages. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but with a check-node complexity the same as binary BP. As every message is a scalar, the message normalization can be naturally applied to improve the performance. Another observation is that the message-update schedule affects the BP decoding performance against short cycles. We show that running BP with message normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error-floor in computer simulation., Comment: to be published in Proc. IEEE Global Commun. Conf. (GLOBECOM), 2020

Quantum error correction is necessary to protect logical quantum states and operations. However, no meaningful data protection can be made when the syndrome extraction is erroneous due to faulty measurement gates. Quantum data-syndrome (DS) codes are designed to protect the data qubits and syndrome bits concurrently. In this paper, we propose an efficient decoding algorithm for quantum DS codes with sparse check matrices. Based on a refined belief propagation (BP) decoding for stabilizer codes, we propose a DS-BP algorithm to handle the quaternary quantum data errors and binary syndrome bit errors. Moreover, a sparse quantum code may inherently be able to handle minor syndrome errors so that fewer redundant syndrome measurements are necessary. We demonstrate this with simulations on a quantum hypergraph-product code.

We describe explicitly the chamber structure of the movable cone for a general smooth complete intersection Calabi-Yau threefold $X$ of Picard number two in certain Pr-ruled Fano manifold and hence verify the Morrison-Kawamata cone conjecture for such $X$. Moreover, all birational minimal models of such Calabi-Yau threefolds are found, whose number is finite up to isomorphism., Comment: 42 pages. Correct some typos in this new version. Comments are welcome

We investigate CO$_2$-driven diffusiophoresis of colloidal particles and bacterial cells in a Hele-Shaw geometry. Combining experiments and a model, we understand the characteristic length and time scales of CO$_2$-driven diffusiophoresis in relation to system dimensions and CO$_2$ diffusivity. Directional migration of wild-type V. cholerae and a mutant lacking flagella, as well as S. aureus and P. aeruginosa, near a dissolving CO$_2$ source shows that diffusiophoresis of bacteria is achieved independent of cell shape and Gram stain. Long-time experiments suggest possible applications for bacterial diffusiophoresis to cleaning systems or anti-biofouling surfaces.

Fault-tolerant quantum computation (FTQC) schemes using large block codes that encode $k>1$ qubits in $n$ physical qubits can potentially reduce the resource overhead to a great extent because of their high encoding rate. However, the fault-tolerant (FT) logical operations for the encoded qubits are difficult to find and implement, which usually takes not only a very large resource overhead but also long $\textit{in-situ}$ computation time. In this paper, we focus on Calderbank-Shor-Steane $[\![ n,k,d ]\!]$ (CSS) codes and their logical FT Clifford circuits. We show that the depth of an arbitrary logical Clifford circuit can be implemented fault-tolerantly in $O(1)$ steps \emph{in-situ} via either Knill or Steane syndrome measurement circuit, with the qualified ancilla states efficiently prepared. Particularly, for those codes satisfying $k/n\sim \Theta(1)$, the resource scaling for Clifford circuits implementation on the logical level can be the same as on the physical level up to a constant, which is independent of code distance $d$. With a suitable pipeline to produce ancilla states, our scheme requires only a modest resource cost in physical qubits, physical gates, and computation time for very large scale FTQC., Comment: 13 pages, 6 figures. This paper supersedes the previous arXiv:1805.12082 by authors; v2:published version

Theoretically speaking, a photon can travel arbitrarily long before it enters into a detector, resulting a click. How much information can a photon carry? We study a bipartite asymmetric "two-way signaling" protocol as an extension of that proposed by Del Santo and Daki\'{c}. Suppose that Alice and Bob are distant from each other and each of them has an $n$-bit string. They are tasked to exchange the information of their local n-bit strings with each other, using only a single photon during the communication. It has been shown that the superposition of different spatial locations in a Mach-Zehnder (MZ) interferometer enables bipartite local encodings. We show that, after the travel of a photon through a cascade of $n$-level MZ interferometers in our protocol, the one of Alice or Bob whose detector clicks can access the other's full information of $n$-bit string, while the other can gain one-bit of information. That is, the wave-particle duality makes two-way signaling possible, and a single photon can carry arbitrarily large (but finite) information., Comment: 6 pages, 6 figures

Quantum stabilizer codes constructed from sparse matrices have good performance and can be efficiently decoded by belief propagation (BP). A conventional BP decoding algorithm treats binary stabilizer codes as additive codes over GF(4). This algorithm has a relatively complex process of handling check-node messages, which incurs higher decoding complexity. Moreover, BP decoding of a stabilizer code usually suffers a performance loss due to the many short cycles in the underlying Tanner graph. In this paper, we propose a refined BP decoding algorithm for quantum codes with complexity roughly the same as binary BP. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but the passed node-to-node messages are single-valued, unlike the quaternary BP, where multivalued node-to-node messages are required. Furthermore, the techniques of message strength normalization can naturally be applied to these single-valued messages to improve the performance. Another observation is that the message-update schedule affects the performance of BP decoding against short cycles. We show that running BP with message strength normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error floor in computer simulation.

Given that approximate quantum error-correcting (AQEC) codes have a potentially better performance than perfect quantum error correction codes, it is pertinent to quantify their performance. While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds do not directly apply to AQEC codes. Herein, we introduce quantum weight enumerators for amplitude damping (AD) errors and work within the framework of approximate quantum error correction. In particular, we introduce an auxiliary exact weight enumerator that is intrinsic to a code space and moreover, we establish a linear relationship between the quantum weight enumerators for AD errors and this auxiliary exact weight enumerator. This allows us to establish a linear program that is infeasible only when AQEC AD codes with corresponding parameters do not exist. To illustrate our linear program, we numerically rule out the existence of three-qubit AD codes that are capable of correcting an arbitrary AD error., Comment: 5 pages

Near-term quantum computers are likely to have small depths due to short coherence time and noisy gates, and thus a potential way to use these quantum devices is using a hybrid scheme that interleaves them with classical computers. For example, the quantum Fourier transform can be implemented by a hybrid of logarithmic-depth quantum circuits and a classical polynomial-time algorithm. Along the line, it seems possible that a general quantum computer may only be polynomially faster than a hybrid quantum-classical computer. Jozsa raised the question of whether $BQP = BPP^{BQNC}$ and conjectured that they are equal, where $BQNC$ means $polylog$-depth quantum circuits. Nevertheless, Aaronson conjectured an oracle separation for these two classes and gave a candidate. In this work, we prove Aaronson's conjecture for a different but related oracle problem. Our result also proves that Jozsa's conjecture fails relative to an oracle.

Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes, including split weight enumerators, generalized MacWilliams identities, and linear programming bounds. In particular, we derive Singleton and Hamming-type upper bounds on degenerate quantum DS codes. Then we study random DS codes and show that random DS codes with a relatively small additional syndrome measurements achieve the Gilbert-Varshamov bound of stabilizer codes. Constructions of quantum DS codes are also discussed. A family of quantum DS codes is based on classical linear block codes, called syndrome measurement codes, so that syndrome bits are encoded in additional redundant stabilizer measurements. Another family of quantum DS codes is CSS-type quantum DS codes based on classical cyclic codes, and this includes the Steane code and the quantum Golay code., Comment: 18 pages, 4 figures. This version comprehensively extends our previous paper (arXiv:1602.01545)

A quantum network, which involves multiple parties pinging each other with quantum messages, could revolutionize communication, computing and basic sciences. The future internet will be a global system of various packet switching quantum and classical networks and we call it \emph{quantum internet}. To build a quantum internet, unified protocols that support the distribution of quantum messages within it are necessary. Intuitively one would extend classical internet protocols to handle quantum messages. However, classical network mechanisms, especially those related to error control and reliable connection, implicitly assume that information can be duplicated, which is not true in the quantum world due to the no-cloning theorem and monogamy of entanglement. In this paper, we investigate and propose protocols for packet quantum network intercommunication. To handle the packet loss problem in transport, we propose a quantum retransmission protocol based on the recursive use of a quantum secret sharing scheme. Other internet protocols are also discussed. In particular, the creation of logical process-to-process connections is accomplished by a quantum version of the three-way handshake protocol., Comment: 8 pages and 4 figures

Quantum error-correcting codes are used to protect quantum information from decoherence. A raw state is mapped, by an encoding circuit, to a codeword so that the most likely quantum errors from a noisy quantum channel can be removed after a decoding process. A good encoding circuit should have some desired features, such as low depth, few gates, and so on. In this paper, we show how to practically implement an encoding circuit of gate complexity $O(n(n-k+c)/\log n)$ for an $[[n,k;c]]$ quantum stabilizer code with the help of $c$ pairs of maximally-entangled states. For the special case of an $[[n,k]]$ stabilizer code with $c=0$, the encoding complexity is $O(n(n-k)/\log n)$, which is previously known to be $O(n^2/\log n)$. For $c>0,$ this suggests that the benefits from shared entanglement come at an additional cost of encoding complexity. Finally we discuss decoding of entanglement-assisted quantum stabilizer codes and extend previously known computational hardness results on decoding quantum stabilizer codes., Comment: accepted by the 2019 IEEE International Symposium on Information Theory (ISIT2019)

In (single-server) Private Information Retrieval (PIR), a server holds a large database $DB$ of size $n$, and a client holds an index $i \in [n]$ and wishes to retrieve $DB[i]$ without revealing $i$ to the server. It is well known that information theoretic privacy even against an `honest but curious' server requires $\Omega(n)$ communication complexity. This is true even if quantum communication is allowed and is due to the ability of such an adversarial server to execute the protocol on a superposition of databases instead of on a specific database (`input purification attack'). Nevertheless, there have been some proposals of protocols that achieve sub-linear communication and appear to provide some notion of privacy. Most notably, a protocol due to Le Gall (ToC 2012) with communication complexity $O(\sqrt{n})$, and a protocol by Kerenidis et al. (QIC 2016) with communication complexity $O(\log(n))$, and $O(n)$ shared entanglement. We show that, in a sense, input purification is the only potent adversarial strategy, and protocols such as the two protocols above are secure in a restricted variant of the quantum honest but curious (a.k.a specious) model. More explicitly, we propose a restricted privacy notion called \emph{anchored privacy}, where the adversary is forced to execute on a classical database (i.e. the execution is anchored to a classical database). We show that for measurement-free protocols, anchored security against honest adversarial servers implies anchored privacy even against specious adversaries. Finally, we prove that even with (unlimited) pre-shared entanglement it is impossible to achieve security in the standard specious model with sub-linear communication, thus further substantiating the necessity of our relaxation. This lower bound may be of independent interest (in particular recalling that PIR is a special case of Fully Homomorphic Encryption).

The leakage chain rule for quantum min-entropy quantifies the change of min-entropy when one party gets additional leakage about the information source. Herein we provide an interactive version that quantifies the change of min-entropy between two parties, who share an initial classical-quantum state and are allowed to run a two-party protocol. As an application, we prove new versions of lower bounds on the complexity of quantum communication of classical information., Comment: A few terminology mistakes were corrected in this version

Clifford circuits play an important role in quantum computation. Gottesman and Chuang proposed a gate teleportation protocol so that a quantum circuit can be implemented by the teleportation circuit with specific ancillary qubits. In particular, an $n$-qubit Clifford circuit $U$ can be implemented by preparing an ancillary stabilizer state $(I\otimes U)|\Phi^+\rangle^{\otimes n}$ for teleportation and doing a Pauli correction conditioned on the measurement. In this paper, we provide an alternative procedure to implement a Clifford circuit through Pauli measurements, by preparing $O(1)$ ancillas that are Calderbank-Shor-Steane (CSS) stabilizer states. That is to say, $O(1)$ CSS states are sufficient to implement any Clifford circuit. As an application to fault-tolerant quantum computation, any Clifford circuit can be implemented by $O(1)$ steps of Steane syndrome extraction if clean CSS stabilizer states are available., Comment: 9 pages, comments are most welcome

The famous Shannon impossibility result says that any encryption scheme with perfect secrecy requires a secret key at least as long as the message. In this paper we provide its quantum analogue with imperfect secrecy and imperfect correctness. We also give a systematic study of information-theoretically secure quantum encryption with two secrecy definitions. We show that the weaker one implies the stronger but with a security loss in $d$, where $d$ is the dimension of the encrypted quantum system. This is good enough if the target secrecy error is of $o(d^{-1})$., Comment: 10 pages. corrected a few errors in the previous version