arXiv subjects

Peter W. Shor

Publications and source records attributed to Peter W. Shor.

At least 19 recordsLinked to original sources

arXiv2025

Universal graph representation of stabilizer codes

While stabilizer tableaus have proven useful as a descriptive tool for additive quantum codes, they otherwise offer little guidance for concrete constructions or algorithm analysis. We introduce a representation of stabilizer codes as graphs with certain structures, and prove via the ZX Calculus that this representation is related to stabilizer tableaus by an efficiently computable bijection. This gives a new universal recipe for code construction by way of finding graphs with nice properties. The graph representation gives insight into both code construction and algorithms. We construct as examples families of $[[ n, \;Θ(\frac{n}{\log n}), \;Θ(\log n)]]$ and $[[ n, \;Ω(n^{4/5}), \;Θ(n^{1/5}) ]]$ codes. We use graphs in a probabilistic analysis to extend the quantum Gilbert-Varshamov bound into a three-way distance-rate-weight trade-off. Moreover, code properties such as distance and encoding circuit depth are bounded by simple functions of the graph degree. We prove that key coding algorithms -- distance approximation, minimum weight generator selection, and decoding -- are unified as instances of one optimization game on a graph. By studying this game, we construct an efficient greedy decoder and prove that it corrects all recoverable errors for all graphs with cycle lengths no shorter than 13 (reducible to 5 with mild extra constraints); these include the above two families. Our results suggest that graphs are generically useful for the study of stabilizer codes.

quant-ph↗

arXiv2025

Graphical quantum Clifford-encoder compilers from the ZX calculus

We present a quantum compilation algorithm that maps Clifford encoders, encoding maps for stabilizer quantum codes, to a unique graphical representation in the ZX calculus. Specifically, we develop a canonical form in the ZX calculus and prove canonicity as well as efficient reducibility of any Clifford encoder into the canonical form. The diagrams produced by our compiler visualize information propagation and entanglement structure of the encoder, revealing properties that may be obscured in the circuit or stabilizer-tableau representation. Consequently, our canonical representation may be an informative technique for the design of new stabilizer quantum codes via graph theory analysis.

quant-ph↗

arXiv2024

A New Class of Algorithms for Finding Short Vectors in Lattices Lifted from Co-dimension $k$ Codes

We introduce a new class of algorithms for finding a short vector in lattices defined by codes of co-dimension $k$ over $\mathbb{Z}_P^d$, where $P$ is prime. The co-dimension $1$ case is solved by exploiting the packing properties of the projections mod $P$ of an initial set of non-lattice vectors onto a single dual codeword. The technical tools we introduce are sorting of the projections followed by single-step pairwise Euclidean reduction of the projections, resulting in monotonic convergence of the positive-valued projections to zero. The length of vectors grows by a geometric factor each iteration. For fixed $P$ and $d$, and large enough user-defined input sets, we show that it is possible to minimize the number of iterations, and thus the overall length expansion factor, to obtain a short lattice vector. Thus we obtain a novel approach for controlling the output length, which resolves an open problem posed by Noah Stephens-Davidowitz (the possibility of an approximation scheme for the shortest-vector problem (SVP) which does not reduce to near-exact SVP). In our approach, one may obtain short vectors even when the lattice dimension is quite large, e.g., 8000. For fixed $P$, the algorithm yields shorter vectors for larger $d$. We additionally present a number of extensions and generalizations of our fundamental co-dimension $1$ method. These include a method for obtaining many different lattice vectors by multiplying the dual codeword by an integer and then modding by $P$; a co-dimension $k$ generalization; a large input set generalization; and finally, a "block" generalization, which involves the replacement of pairwise (Euclidean) reduction by a $k$-party (non-Euclidean) reduction. The $k$-block generalization of our algorithm constitutes a class of polynomial-time algorithms indexed by $k\geq 2$, which yield successively improved approximations for the short vector problem.

cs.CR↗

arXiv2023

Bounding the forward classical capacity of bipartite quantum channels

We introduce various measures of forward classical communication for bipartite quantum channels. Since a point-to-point channel is a special case of a bipartite channel, the measures reduce to measures of classical communication for point-to-point channels. As it turns out, these reduced measures have been reported in prior work of Wang et al. on bounding the classical capacity of a quantum channel. As applications, we show that the measures are upper bounds on the forward classical capacity of a bipartite channel. The reduced measures are upper bounds on the classical capacity of a point-to-point quantum channel assisted by a classical feedback channel. Some of the various measures can be computed by semi-definite programming.

quant-ph↗

arXiv2022

The Early Days of Quantum Computation

I recount some of my memories of the early development of quantum computation, including the discovery of the factoring algorithm, of error correcting codes, and of fault tolerance.

quant-ph↗

arXiv2022

Publicly verifiable quantum money from random lattices

Publicly verifiable quantum money is a protocol for the preparation of quantum states that can be efficiently verified by any party for authenticity but is computationally infeasible to counterfeit. We develop a cryptographic scheme for publicly verifiable quantum money based on Gaussian superpositions over random lattices. We introduce a verification-of-authenticity procedure based on the lattice discrete Fourier transform, and subsequently prove the unforgeability of our quantum money under the hardness of the short vector problem from lattice-based cryptography.

quant-ph↗

arXiv2020

A Separation of Out-of-time-ordered Correlation and Entanglement

The out-of-time-ordered correlation (OTOC) and entanglement are two physically motivated and widely used probes of the "scrambling" of quantum information, a phenomenon that has drawn great interest recently in quantum gravity and many-body physics. We argue that the corresponding notions of scrambling can be fundamentally different, by proving an asymptotic separation between the time scales of the saturation of OTOC and that of entanglement entropy in a random quantum circuit model defined on graphs with a tight bottleneck, such as tree graphs. Our result counters the intuition that a random quantum circuit mixes in time proportional to the diameter of the underlying graph of interactions. It also provides a more rigorous justification for an argument in our previous work arXiv:1807.04363, that black holes may be slow information scramblers, which in turn relates to the black hole information problem. The bounds we obtained for OTOC are interesting in their own right in that they generalize previous studies of OTOC on lattices to the geometries on graphs in a rigorous and general fashion.

quant-ph↗

arXiv2019

Entropy Bound for the Classical Capacity of a Quantum Channel Assisted by Classical Feedback

We prove that the classical capacity of an arbitrary quantum channel assisted by a free classical feedback channel is bounded from above by the maximum average output entropy of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve the classical capacity of a quantum erasure channel, and by taking into account energy constraints, we conclude the same for a pure-loss bosonic channel. The method for establishing the aforementioned entropy bound involves identifying an information measure having two key properties: 1) it does not increase under a one-way local operations and classical communication channel from the receiver to the sender and 2) a quantum channel from sender to receiver cannot increase the information measure by more than the maximum output entropy of the channel. This information measure can be understood as the sum of two terms, with one corresponding to classical correlation and the other to entanglement.

quant-ph↗

arXiv2019

The Quantum Double Model with Boundary: Condensations and Symmetries

Associated to every finite group, Kitaev has defined the quantum double model for every orientable surface without boundary. In this paper, we define boundaries for this model and characterize condensations; that is, we find all quasi-particle excitations (anyons) which disappear when they move to the boundary. We then consider two phases of the quantum double model corresponding to two groups with a domain wall between them, and study the tunneling of anyons from one phase to the other. Using this framework we discuss the necessary and sufficient conditions when two different groups give the same anyon types. As an application we show that in the quantum double model for S_3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. This group is indeed a special case of groups of the form of the semidirect product of the additive and multiplicative groups of a finite field, for all of which we prove a similar symmetry.

quant-ph↗

arXiv2018

Scrambling Time and Causal Structure of the Photon Sphere of a Schwarzschild Black Hole

Recently, physicists have started applying quantum information theory to black holes. This led to the conjecture that black holes are the fastest scramblers of information, and that they scramble it in time order M log M, where M is the mass of the black hole in natural units. As stated above, the conjecture is not completely defined, as there are several possible definitions of scrambling times. It appears that not all papers that refer to this conjecture interpret it the same way. We consider a definition of scrambling time stronger than the one given in the paper that first proposed this conjecture [Sekino and Susskind, JHEP 0810:065 (2008)], and show that this stronger version of the conjecture appears to be incompatible with a number of other widely-believed and reasonable-sounding properties of black holes. We argue that for the scrambling time of a black hole to be this fast, either relativity is violated or non-standard physics must be occurring outside the stretched event horizon of a black hole. More specifically, either information is being transferred faster than relativity would permit, the information is not carried by the Hawking radiation and thus must be carried by unknown physics, or the Hawking radiation carries much more information than standard thermodynamics would permit.

gr-qc↗

arXiv2018

Resource theory of non-Gaussian operations

Non-Gaussian states and operations are crucial for various continuous-variable quantum information processing tasks. To quantitatively understand non-Gaussianity beyond states, we establish a resource theory for non-Gaussian operations. In our framework, we consider Gaussian operations as free operations, and non-Gaussian operations as resources. We define entanglement-assisted non-Gaussianity generating power and show that it is a monotone that is non-increasing under the set of free super-operations, i.e., concatenation and tensoring with Gaussian channels. For conditional unitary maps, this monotone can be analytically calculated. As examples, we show that the non-Gaussianity of ideal photon-number subtraction and photon-number addition equal the non-Gaussianity of the single-photon Fock state. Based on our non-Gaussianity monotone, we divide non-Gaussian operations into two classes: (1) the finite non-Gaussianity class, e.g., photon-number subtraction, photon-number addition and all Gaussian-dilatable non-Gaussian channels; and (2) the diverging non-Gaussianity class, e.g., the binary phase-shift channel and the Kerr nonlinearity. This classification also implies that not all non-Gaussian channels are exactly Gaussian-dilatable. Our resource theory enables a quantitative characterization and a first classification of non-Gaussian operations, paving the way towards the full understanding of non-Gaussianity.

quant-ph↗

arXiv2017

Superadditivity in trade-off capacities of quantum channels

In this article, we investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading classical communication, quantum communication and entanglement. Understanding such additivity property is important if we want to optimally use a quantum channel for general communication purpose. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity, and it is largely unknown if this could lead to an superadditive double or triple resource capacity. For example, if a channel has an additive classical and quantum capacity, can the classical-quantum capacity be superadditive? In this work, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple capacity region.

quant-ph↗

arXiv2017

Superadditivity of the Classical Capacity with Limited Entanglement Assistance

Finding the optimal encoding strategies can be challenging for communication using quantum channels, as classical and quantum capacities may be superadditive. Entanglement assistance can often simplify this task, as the entanglement-assisted classical capacity for any channel is additive, making entanglement across channel uses unnecessary. If the entanglement assistance is limited, the picture is much more unclear. Suppose the classical capacity is superadditive, then the classical capacity with limited entanglement assistance could retain superadditivity by continuity arguments. If the classical capacity is additive, it is unknown if superadditivity can still be developed with limited entanglement assistance. We show this is possible, by providing an example. We construct a channel for which, the classical capacity is additive, but that with limited entanglement assistance can be superadditive. This shows entanglement plays a weird role in communication and we still understand very little about it.

quant-ph↗

arXiv2017

Additive Classical Capacity of Quantum Channels Assisted by Noisy Entanglement

We give a capacity formula for the classical information transmission over a noisy quantum channel, with separable encoding by the sender and limited resources provided by the receiver's pre-shared ancilla. Instead of a pure state, we consider the signal-ancilla pair in a mixed state, purified by a "witness". Thus, the signal-witness correlation limits the resource available from the signal-ancilla correlation. Our formula characterizes the utility of different forms of resources, including noisy or limited entanglement assistance, for classical communication. With separable encoding, the sender's signals across multiple channel uses are still allowed to be entangled, yet our capacity formula is additive. In particular, for generalized covariant channels our capacity formula has a simple closed-form. Moreover, our additive capacity formula upper bounds the general coherent attack's information gain in various two-way quantum key distribution protocols. For Gaussian protocols, the additivity of the formula indicates that the collective Gaussian attack is the most powerful.

quant-ph↗

arXiv2016

An Efficient Quantum Algorithm for a Variant of the Closest Lattice-Vector Problem

The Systematic Normal Form (SysNF) is a canonical form of lattices introduced in [Eldar,Shor '16], in which the basis entries satisfy a certain co-primality condition. Using a "smooth" analysis of lattices by SysNF lattices we design a quantum algorithm that can efficiently solve the following variant of the bounded-distance-decoding problem: given a lattice L, a vector v, and numbers b = λ_1(L)/n^{17}, a = λ_1(L)/n^{13} decide if v's distance from L is in the range [a/2, a] or at most b, where λ_1(L) is the length of L's shortest non-zero vector. Improving these parameters to a = b = λ_1(L)/\sqrt{n} would invalidate one of the security assumptions of the Learning-with-Errors (LWE) cryptosystem against quantum attacks.

quant-ph↗

arXiv2016

Power law violation of the area law in quantum spin chains

The sub-volume scaling of the entanglement entropy with the system's size, $n$, has been a subject of vigorous study in the last decade [1]. The area law provably holds for gapped one dimensional systems [2] and it was believed to be violated by at most a factor of $\log\left(n\right)$ in physically reasonable models such as critical systems. In this paper, we generalize the spin$-1$ model of Bravyi et al [3] to all integer spin-$s$ chains, whereby we introduce a class of exactly solvable models that are physical and exhibit signatures of criticality, yet violate the area law by a power law. The proposed Hamiltonian is local and translationally invariant in the bulk. We prove that it is frustration free and has a unique ground state. Moreover, we prove that the energy gap scales as $n^{-c}$, where using the theory of Brownian excursions, we prove $c\ge2$. This rules out the possibility of these models being described by a conformal field theory. We analytically show that the Schmidt rank grows exponentially with $n$ and that the half-chain entanglement entropy to the leading order scales as $\sqrt{n}$ (Eq. 16). Geometrically, the ground state is seen as a uniform superposition of all $s-$colored Motzkin walks. Lastly, we introduce an external field which allows us to remove the boundary terms yet retain the desired properties of the model. Our techniques for obtaining the asymptotic form of the entanglement entropy, the gap upper bound and the self-contained expositions of the combinatorial techniques, more akin to lattice paths, may be of independent interest.

quant-ph↗

arXiv2016

The Systematic Normal Form of Lattices

We introduce a new canonical form of lattices called the systematic normal form (SNF). We show that for every lattice there is an efficiently computable "nearby" SNF lattice, such that for any lattice one can solve lattice problems on its "nearby" SNF lattice, and translate the solutions back efficiently to the original lattice. The SNF provides direct connections between arbitrary lattices, and various lattice related problems like the Shortest-Integer-Solution, Approximate Greatest Common Divisor. As our main application of SNF we derive a new set of worst-to-average case lattice reductions that deviate significantly from the template of Ajtai and improve upon previous reductions in terms of simplicity.

cs.CC↗

arXiv2014

Information Causality, Szemerédi-Trotter and Algebraic Variants of CHSH

In this work, we consider the following family of two prover one-round games. In the CHSH_q game, two parties are given x,y in F_q uniformly at random, and each must produce an output a,b in F_q without communicating with the other. The players' objective is to maximize the probability that their outputs satisfy a+b=xy in F_q. This game was introduced by Buhrman and Massar (PRA 2005) as a large alphabet generalization of the celebrated CHSH game---which is one of the most well-studied two-prover games in quantum information theory, and which has a large number of applications to quantum cryptography and quantum complexity. Our main contributions in this paper are the first asymptotic and explicit bounds on the entangled and classical values of CHSH_q, and the realization of a rather surprising connection between CHSH_q and geometric incidence theory. On the way to these results, we also resolve a problem of Pawlowski and Winter about pairwise independent Information Causality, which, beside being interesting on its own, gives as an application a short proof of our upper bound for the entangled value of CHSH_q.

quant-ph↗