arXiv subjects

Peter Shor

Publications and source records attributed to Peter Shor.

At least 19 recordsLinked to original sources

arXiv2024

The Learning Stabilizers with Noise problem

Random classical codes have good error correcting properties, and yet they are notoriously hard to decode in practice. Despite many decades of extensive study, the fastest known algorithms still run in exponential time. The Learning Parity with Noise (LPN) problem, which can be seen as the task of decoding a random linear code in the presence of noise, has thus emerged as a prominent hardness assumption with numerous applications in both cryptography and learning theory. Is there a natural quantum analog of the LPN problem? In this work, we introduce the Learning Stabilizers with Noise (LSN) problem, the task of decoding a random stabilizer code in the presence of local depolarizing noise. We give both polynomial-time and exponential-time quantum algorithms for solving LSN in various depolarizing noise regimes, ranging from extremely low noise, to low constant noise rates, and even higher noise rates up to a threshold. Next, we provide concrete evidence that LSN is hard. First, we show that LSN includes LPN as a special case, which suggests that it is at least as hard as its classical counterpart. Second, we prove a worst-case to average-case reduction for variants of LSN. We then ask: what is the computational complexity of solving LSN? Because the task features quantum inputs, its complexity cannot be characterized by traditional complexity classes. Instead, we show that the LSN problem lies in a recently introduced (distributional and oracle) unitary synthesis class. Finally, we identify several applications of our LSN assumption, ranging from the construction of quantum bit commitment schemes to the computational limitations of learning from quantum data.

quant-ph↗

arXiv2023

Topological quantum computation assisted by phase transitions

In this paper, we explore topological quantum computation augmented by subphases and phase transitions. We commence by investigating the anyon tunneling map, denoted as $\varphi$, between subphases of the quantum double model $\mathcal{D}(G)$ for any arbitrary finite group $G$. Subsequently, we delve into the relationship between $\varphi$ and the Floquet code, and extend the Abelian Floquet code to encompass non-abelian cases. We conclude by demonstrating how phase transitions in both the temporal and spatial directions can enhance the diversity of topological gates for general topological orders described by modular tensor categories.

quant-ph↗

arXiv2022

Simultaneous Measurement and Entanglement

We study scenarios which arise when two spatially-separated observers, Alice and Bob, are try to identify a quantum state sampled from several possibilities. In particular, we examine their strategies for maximizing both the probability of guessing their state correctly as well as their information gain about it. It is known that there are scenarios where allowing Alice and Bob to use LOCC offers an improvement over the case where they must make their measurements simultaneously. Similarly, Alice and Bob can sometimes improve their outcomes if they have access to a Bell pair. We show how LOCC allows Alice and Bob to distinguish between two product states optimally and find that a LOCC is almost always more helpful than a Bell pair for distinguishing product states.

quant-ph↗

arXiv2017

When is a bit worth much more than kT ln2?

Physical processes thatobtain, process, and erase information involve tradeoffs between information and energy. The fundamental energetic value of a bit of information exchanged with a reservoir at temperature T is kT ln2. This paper investigates the situation in which information is missing about just what physical process is about to take place. The fundamental energetic value of such information can be far greater than kT ln2 per bit.

cond-mat.stat-mech↗

arXiv2017

Maximizing free energy gain

Maximizing the amount of work harvested from an environment is important for a wide variety of biological and technological processes, from energy-harvesting processes such as photosynthesisto energy storage systems such as fuels and batteries. Here we consider the maximization of free energy -- and by extension, the maximum extractable work -- that can be gained by a classical or quantum system that undergoes driving by its environment. We consider how the free energy gain depends on the initial state of the system, while also accounting for the cost of preparing the system. We provide simple necessary and sufficient conditions for increasing the gain of free energy by varying the initial state. We also derive simple formulae that relate the free energy gained using the optimal initial state rather than another suboptimal initial state. Finally, we demonstrate that the problem of finding the optimal initial state may have two distinct regimes, one easy and one difficult, depending on the temperatures used for preparation and work extraction. We illustrate our results on a simple model of an information engine.

cond-mat.stat-mech↗

arXiv2017

A Discrete Fourier Transform on Lattices with Quantum Applications

In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in $\R^n$, that generalizes the $n$-th fold DFT of the integer lattice $\Z^n$ to arbitrary lattices. This definition is not applicable for every lattice, but can be defined on lattices known as Systematic Normal Form (SysNF) introduced in \cite{ES16}. Systematic Normal Form lattices are sets of integer vectors that satisfy a single homogeneous modular equation, which itself satisfies a certain number-theoretic property. Such lattices form a dense set in the space of $n$-dimensional lattices, and can be used to approximate efficiently any lattice. This implies that for every lattice $L$ a DFT can be computed efficiently on a lattice near $L$. Our proof of the statement above uses arguments from quantum computing, and as an application of our definition we show a quantum algorithm for sampling from discrete distributions on lattices, that extends our ability to sample efficiently from the discrete Gaussian distribution \cite{GPV08} to any distribution that is sufficiently "smooth". We conjecture that studying the eigenvectors of the newly-defined lattice DFT may provide new insights into the structure of lattices, especially regarding hard computational problems, like the shortest vector problem.

quant-ph↗

arXiv2017

polylog-LDPC Capacity Achieving Codes for the Noisy Quantum Erasure Channel

We provide $poly\log$ sparse quantum codes for correcting the erasure channel arbitrarily close to the capacity. Specifically, we provide $[[n, k, d]]$ quantum stabilizer codes that correct for the erasure channel arbitrarily close to the capacity if the erasure probability is at least $0.33$, and with a generating set $\langle S_1, S_2, ... S_{n-k} \rangle$ such that $|S_i|\leq \log^{2+\zeta}(n)$ for all $i$ and for any $\zeta > 0$ with high probability. In this work we show that the result of Delfosse et al. is tight: one can construct capacity approaching codes with weight almost $O(1)$.

quant-ph↗

arXiv2017

Quantum and Super-quantum enhancements to two-sender, two-receiver channels

We study the consequences of 'super-quantum non-local correlations' as represented by the PR-box model of Popescu and Rohrlich, and show PR-boxes can enhance the capacity of noisy interference channels between two senders and two receivers. PR-box correlations violate Bell/CHSH inequalities and are thus stronger -- more non-local -- than quantum mechanics; yet weak enough to respect special relativity in prohibiting faster-than-light communication. Understanding their power will yield insight into the non-locality of quantum mechanics. We exhibit two proof-of-concept channels: first, we show a channel between two sender-receiver pairs where the senders are not allowed to communicate, for which a shared super-quantum bit (a PR-box) allows perfect communication. This feat is not achievable with the best classical (senders share no resources) or quantum entanglement-assisted (senders share entanglement) strategies. Second, we demonstrate a class of channels for which a tunable parameter achieves a double separation of capacities; for some range of \epsilon, the super-quantum assisted strategy does better than the entanglement-assisted strategy, which in turn does better than the classical one.

quant-ph↗

arXiv2015

Time Independent Universal Computing with Spin Chains: Quantum Plinko Machine

We present a scheme for universal quantum computing using XY Heisenberg spin chains. Information is encoded into packets propagating down these chains, and they interact with each other to perform universal quantum computation. A circuit using g gate blocks on m qubits can be encoded into chains of length $O(g^{3+\delta} m^{3+\delta})$ for all $\delta >0$ with vanishingly small error.

quant-ph↗

arXiv2015

Efficiently Controllable Graphs

We investigate graphs that can be disconnected into small components by removing a vanishingly small fraction of their vertices. We show that when a quantum network is described by such a graph, the network is efficiently controllable, in the sense that universal quantum computation can be performed using a control sequence polynomial in the size of the network while controlling a vanishingly small fraction of subsystems. We show that networks corresponding to finite-dimensional lattices are efficently controllable, and explore generalizations to percolation clusters and random graphs. We show that the classical computational complexity of estimating the ground state of Hamiltonians described by controllable graphs is polynomial in the number of subsystems/qubits.

quant-ph↗

arXiv2014

Different Strategies for Optimization Using the Quantum Adiabatic Algorithm

We present the results of a numerical study, with 20 qubits, of the performance of the Quantum Adiabatic Algorithm on randomly generated instances of MAX 2-SAT with a unique assignment that maximizes the number of satisfied clauses. The probability of obtaining this assignment at the end of the quantum evolution measures the success of the algorithm. Here we report three strategies which consistently increase the success probability for the hardest instances in our ensemble: decreasing the overall evolution time, initializing the system in excited states, and adding a random local Hamiltonian to the middle of the evolution.

quant-ph↗

arXiv2013

New Constructions of Codes for Asymmetric Channels via Concatenation

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov-Tenengol'ts-Constantin-Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. In the binary case, Varshamov has shown that almost all good linear codes for the asymmetric channel are also good for the symmetric channel. Our results indicate that Varshamov's argument does not extend to the nonbinary case, i.e., one can find better linear codes for asymmetric channels than for symmetric ones.

cs.IT↗

arXiv2012

The performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs

In this paper we study the performance of the quantum adiabatic algorithm on random instances of two combinatorial optimization problems, 3-regular 3-XORSAT and 3-regular Max-Cut. The cost functions associated with these two clause-based optimization problems are similar as they are both defined on 3-regular hypergraphs. For 3-regular 3-XORSAT the clauses contain three variables and for 3-regular Max-Cut the clauses contain two variables. The quantum adiabatic algorithms we study for these two problems use interpolating Hamiltonians which are stoquastic and therefore amenable to sign-problem free quantum Monte Carlo and quantum cavity methods. Using these techniques we find that the quantum adiabatic algorithm fails to solve either of these problems efficiently, although for different reasons.

quant-ph↗

arXiv2012

Criticality without frustration for quantum spin-1 chains

Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s=1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right parentheses separated by empty spaces. Entanglement entropy of one half of the chain scales as log(n)/2 + O(1), where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1/n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.

quant-ph↗

arXiv2011

Quantum state restoration and single-copy tomography

Given a single copy of an n qubit quantum state |psi>, the no-cloning theorem greatly limits the amount of information which can be extracted from it. Moreover, given only a procedure which verifies the state, for example a procedure which measures the operator |psi> in time polynomial in n . In this paper, we consider the scenario in which we are given both a single copy of |psi> and the ability to verify it. We show that in this setting, we can do several novel things efficiently. We present a new algorithm that we call quantum state restoration which allows us to extend a large subsystem of |psi> to the full state, and in turn this allows us to copy small subsystems of |psi>. In addition, we present algorithms that can perform tomography on small subsystems of |psi>, and we show how to use these algorithms to estimate the statistics of any efficiently implementable POVM acting on |psi> in time polynomial in the number of outcomes of the POVM.

quant-ph↗

arXiv2010

Unstructured Randomness, Small Gaps and Localization

We study the Hamiltonian associated with the quantum adiabatic algorithm with a random cost function. Because the cost function lacks structure we can prove results about the ground state. We find the ground state energy as the number of bits goes to infinity, show that the minimum gap goes to zero exponentially quickly, and we see a localization transition. We prove that there are no levels approaching the ground state near the end of the evolution. We do not know which features of this model are shared by a quantum adiabatic algorithm applied to random instances of satisfiability since despite being random they do have bit structure.

quant-ph↗

arXiv2010

Quantum Adiabatic Algorithms, Small Gaps, and Different Paths

We construct a set of instances of 3SAT which are not solved efficiently using the simplest quantum adiabatic algorithm. These instances are obtained by picking random clauses all consistent with two disparate planted solutions and then penalizing one of them with a single additional clause. We argue that by randomly modifying the beginning Hamiltonian, one obtains (with substantial probability) an adiabatic path that removes this difficulty. This suggests that the quantum adiabatic algorithm should in general be run on each instance with many different random paths leading to the problem Hamiltonian. We do not know whether this trick will help for a random instance of 3SAT (as opposed to an instance from the particular set we consider), especially if the instance has an exponential number of disparate assignments that violate few clauses. We use a continuous imaginary time Quantum Monte Carlo algorithm in a novel way to numerically investigate the ground state as well as the first excited state of our system. Our arguments are supplemented by Quantum Monte Carlo data from simulations with up to 150 spins.

quant-ph↗

arXiv2010

Quantum money from knots

Quantum money is a cryptographic protocol in which a mint can produce a quantum state, no one else can copy the state, and anyone (with a quantum computer) can verify that the state came from the mint. We present a concrete quantum money scheme based on superpositions of diagrams that encode oriented links with the same Alexander polynomial. We expect our scheme to be secure against computationally bounded adversaries.

quant-ph↗