arXiv subjects

Ashwin Nayak

Publications and source records attributed to Ashwin Nayak.

At least 37 records · Page 2Linked to original sources

arXiv2020

Capacity Approaching Coding for Low Noise Interactive Quantum Communication, Part I: Large Alphabets

We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with $n$ messages, designed for a noiseless qudit channel over a $\mathrm{poly}(n)$ size alphabet, our main result is a simulation method that fails with probability less than $2^{-Θ(nε)}$ and uses a qudit channel over the same alphabet $n\left(1+Θ\left(\sqrtε\right)\right)$ times, of which an $ε$ fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the $\sqrtε$ term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Our work improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCS'14] for low $ε$.

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arXiv2019

Online Learning of Quantum States

Suppose we have many copies of an unknown $n$-qubit state $ρ$. We measure some copies of $ρ$ using a known two-outcome measurement $E_{1}$, then other copies using a measurement $E_{2}$, and so on. At each stage $t$, we generate a current hypothesis $σ_{t}$ about the state $ρ$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\operatorname{Tr}(E_{i} σ_{t}) - \operatorname{Tr}(E_{i}ρ) |$, the error in our prediction for the next measurement, is at least $\varepsilon$ at most $\operatorname{O}\!\left(n / \varepsilon^2 \right) $ times. Even in the "non-realizable" setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that do significantly worse than the best possible states at most $\operatorname{O}\!\left(\sqrt {Tn}\right) $ times on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.

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arXiv2019

Noisy Interactive Quantum Communication

We study the problem of simulating protocols in a quantum communication setting over noisy channels. This problem falls at the intersection of quantum information theory and quantum communication complexity, and it will be of importance for eventual real-world applications of interactive quantum protocols, which can be proved to have exponentially lower communication costs than their classical counterparts for some problems. These are the first results concerning the quantum version of this problem, originally studied by Schulman in a classical setting (FOCS '92, STOC '93). We simulate a length $N$ quantum communication protocol by a length $O(N)$ protocol with arbitrarily small error. Under adversarial noise, our strategy can withstand, for arbitrarily small $ε> 0$, error rates as high as $1/2 -ε$ when parties pre-share perfect entanglement, but the classical channel is noisy. We show that this is optimal. We provide extension of these results in several other models of communication, including when also the entanglement is noisy, and when there is no pre-shared entanglement but communication is quantum and noisy. We also study the case of random noise, for which we provide simulation protocols with positive communication rates and no pre-shared entanglement over some quantum channels with quantum capacity $C_Q=0$, proving that $C_Q$ is in general not the right characterization of a channel's capacity for interactive quantum communication. Our results are stated for a general quantum communication protocol in which Alice and Bob collaborate, and these results hold in particular in the quantum communication complexity settings of the Yao and Cleve--Buhrman models.

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arXiv2018

Communication Complexity of One-Shot Remote State Preparation

Quantum teleportation uses prior shared entanglement and classical communication to send an unknown quantum state from one party to another. Remote state preparation (RSP) is a similar distributed task in which the sender knows the entire classical description of the state to be sent. (This may also be viewed as the task of non-oblivious compression of a single sample from an ensemble of quantum states.) We study the communication complexity of approximate remote state preparation, in which the goal is to prepare an approximation of the desired quantum state. Jain [Quant. Inf. & Comp., 2006] showed that the worst-case communication complexity of approximate RSP can be bounded from above in terms of the maximum possible information in an encoding. He also showed that this quantity is a lower bound for communication complexity of (exact) remote state preparation. In this work, we tightly characterize the worst-case and average-case communication complexity of remote state preparation in terms of non-asymptotic information-theoretic quantities. We also show that the average-case communication complexity of RSP can be much smaller than the worst-case one. In the process, we show that n bits cannot be communicated with less than n transmitted bits in LOCC protocols. This strengthens a result due to Nayak and Salzman [J. ACM, 2006] and may be of independent interest.

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arXiv2015

Quantum and classical coin-flipping protocols based on bit-commitment and their point games

We focus on a family of quantum coin-flipping protocols based on bit-commitment. We discuss how the semidefinite programming formulations of cheating strategies can be reduced to optimizing a linear combination of fidelity functions over a polytope. These turn out to be much simpler semidefinite programs which can be modelled using second-order cone programming problems. We then use these simplifications to construct their point games as developed by Kitaev. We also study the classical version of these protocols and use linear optimization to formulate optimal cheating strategies. We then construct the point games for the classical protocols as well using the analysis for the quantum case. We discuss the philosophical connections between the classical and quantum protocols and their point games as viewed from optimization theory. In particular, we observe an analogy between a spectrum of physical theories (from classical to quantum) and a spectrum of convex optimization problems (from linear programming to semidefinite programming, through second-order cone programming). In this analogy, classical systems correspond to linear programming problems and the level of quantum features in the system is correlated to the level of sophistication of the semidefinite programming models on the optimization side. Concerning security analysis, we use the classical point games to prove that every classical protocol of this type allows exactly one of the parties to entirely determine the coin-flip. Using the relationships between the quantum and classical protocols, we show that only "classical" protocols can saturate Kitaev's lower bound for strong coin-flipping. Moreover, if the product of Alice and Bob's optimal cheating probabilities is 1/2, then one party can cheat with probability 1. This rules out quantum protocols of this type from attaining the optimal level of security.

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arXiv2014

The space complexity of recognizing well-parenthesized expressions in the streaming model: the Index function revisited

We show an Omega(sqrt{n}/T) lower bound for the space required by any unidirectional constant-error randomized T-pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well-parenthesized. This proves a conjecture due to Magniez, Mathieu, and Nayak (2009) and rigorously establishes that bidirectional streams are exponentially more efficient in space usage as compared with unidirectional ones. We obtain the lower bound by establishing the minimum amount of information that is necessarily revealed by the players about their respective inputs in a two-party communication protocol for a variant of the Index function, namely Augmented Index. The information cost trade-off is obtained by a novel application of the conceptually simple and familiar ideas such as average encoding and the cut-and-paste property of randomized protocols. Motivated by recent examples of exponential savings in space by streaming quantum algorithms, we also study quantum protocols for Augmented Index. Defining an appropriate notion of information cost for quantum protocols involves a delicate balancing act between its applicability and the ease with which we can analyze it. We define a notion of quantum information cost which reflects some of the non-intuitive properties of quantum information and give a trade-off for this notion. While this trade-off demonstrates the strength of our proof techniques, it does not lead to a space lower bound for checking parentheses. We leave such an implication for quantum streaming algorithms as an intriguing open question.

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arXiv2014

A search for quantum coin-flipping protocols using optimization techniques

Coin-flipping is a cryptographic task in which two physically separated, mistrustful parties wish to generate a fair coin-flip by communicating with each other. Chailloux and Kerenidis (2009) designed quantum protocols that guarantee coin-flips with near optimal bias. The probability of any outcome in these protocols is provably at most $1/\sqrt{2} + δ$ for any given $δ> 0$. However, no explicit description of these protocols is known, and the number of rounds in the protocols tends to infinity as $δ$ goes to 0. In fact, the smallest bias achieved by known explicit protocols is $1/4$ (Ambainis, 2001). We take a computational optimization approach, based mostly on convex optimization, to the search for simple and explicit quantum strong coin-flipping protocols. We present a search algorithm to identify protocols with low bias within a natural class, protocols based on bit-commitment (Nayak and Shor, 2003) restricting to commitment states used by Mochon (2005). An analysis of the resulting protocols via semidefinite programs (SDPs) unveils a simple structure. For example, we show that the SDPs reduce to second-order cone programs. We devise novel cheating strategies in the protocol by restricting the semidefinite programs and use the strategies to prune the search. The techniques we develop enable a computational search for protocols given by a mesh over the parameter space. The protocols have up to six rounds of communication, with messages of varying dimension and include the best known explicit protocol (with bias 1/4). We conduct two kinds of search: one for protocols with bias below 0.2499, and one for protocols in the neighbourhood of protocols with bias 1/4. Neither of these searches yields better bias. Based on the mathematical ideas behind the search algorithm, we prove a lower bound on the bias of a class of four-round protocols.

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arXiv2013

Improved bounds for the randomized decision tree complexity of recursive majority

We consider the randomized decision tree complexity of the recursive 3-majority function. We prove a lower bound of $(1/2-δ) \cdot 2.57143^h$ for the two-sided-error randomized decision tree complexity of evaluating height $h$ formulae with error $δ\in [0,1/2)$. This improves the lower bound of $(1-2δ)(7/3)^h$ given by Jayram, Kumar, and Sivakumar (STOC'03), and the one of $(1-2δ) \cdot 2.55^h$ given by Leonardos (ICALP'13). Second, we improve the upper bound by giving a new zero-error randomized decision tree algorithm that has complexity at most $(1.007) \cdot 2.64944^h$. The previous best known algorithm achieved complexity $(1.004) \cdot 2.65622^h$. The new lower bound follows from a better analysis of the base case of the recursion of Jayram et al. The new algorithm uses a novel "interleaving" of two recursive algorithms.

cs.DS↗

arXiv2012

Short proofs of the Quantum Substate Theorem

The Quantum Substate Theorem due to Jain, Radhakrishnan, and Sen (2002) gives us a powerful operational interpretation of relative entropy, in fact, of the observational divergence of two quantum states, a quantity that is related to their relative entropy. Informally, the theorem states that if the observational divergence between two quantum states rho, sigma is small, then there is a quantum state rho' close to rho in trace distance, such that rho' when scaled down by a small factor becomes a substate of sigma. We present new proofs of this theorem. The resulting statement is optimal up to a constant factor in its dependence on observational divergence. In addition, the proofs are both conceptually simpler and significantly shorter than the earlier proof.

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arXiv2011

Inverting a permutation is as hard as unordered search

We show how an algorithm for the problem of inverting a permutation may be used to design one for the problem of unordered search (with a unique solution). Since there is a straightforward reduction in the reverse direction, the problems are essentially equivalent. The reduction we present helps us bypass the hybrid argument due to Bennett, Bernstein, Brassard, and Vazirani (1997) and the quantum adversary method due to Ambainis (2002) that were earlier used to derive lower bounds on the quantum query complexity of the problem of inverting permutations. It directly implies that the quantum query complexity of the problem is asymptotically the same as that for unordered search, namely in Theta(sqrt(n)).

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arXiv2011

Search via Quantum Walk

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk á la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

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arXiv2009

Interacting boson problems are QMA-hard

Computing the ground-state energy of interacting electron (fermion) problems has recently been shown to be hard for QMA, a quantum analogue of the complexity class NP. Fermionic problems are usually hard, a phenomenon widely attributed to the so-called sign problem occurring in Quantum Monte Carlo simulations. The corresponding bosonic problems are, according to conventional wisdom, tractable. Here, we discuss the complexity of interacting boson problems and show that they are also QMA-hard. In addition, we show that the bosonic version of the so-called N-representability problem is QMA-complete, as hard as its fermionic version. As a consequence, these problems are unlikely to have efficient quantum algorithms.

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arXiv2008

On the hitting times of quantum versus random walks

In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Further, we prove that for any reversible ergodic Markov chain $P$, the quantum hitting time of the quantum analogue of $P$ has the same order as the square root of the classical hitting time of $P$. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. Finally, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems.

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arXiv2007

A Separation between Divergence and Holevo Information for Ensembles

The notion of divergence information of an ensemble of probability distributions was introduced by Jain, Radhakrishnan, and Sen in the context of the ``substate theorem''. Since then, divergence has been recognized as a more natural measure of information in several situations in quantum and classical communication. We construct ensembles of probability distributions for which divergence information may be significantly smaller than the more standard Holevo information. As a result, we establish that lower bounds previously shown for Holevo information are weaker than similar ones shown for divergence information.

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arXiv2007

Accessible versus Holevo Information for a Binary Random Variable

The accessible information acc(E) of an ensemble E is the maximum mutual information between a random variable encoded into quantum states, and the probabilistic outcome of a quantum measurement of the encoding. Accessible information is extremely difficult to characterize analytically; even bounds on it are hard to place. The celebrated Holevo bound states that accessible information cannot exceed chi(E), the quantum mutual information between the random variable and its encoding. However, for general ensembles, the gap between the acc(E) and chi(E) may be arbitrarily large. We consider the special case of a binary random variable, which often serves as a stepping stone towards other results in information theory and communication complexity. We give explicit lower bounds on the the accessible information acc(E) of an ensemble E = {(p, rho_0), (1-p, rho_1)}, with 0 <= p <= 1, as functions of p and chi(E). The bounds are incomparable in the sense that they surpass each other in different parameter regimes. Our bounds arise by measuring the ensemble according to a complete orthogonal measurement that preserves the fidelity of the states rho_0,rho_1. As an intermediate step, therefore, we give new relations between the two quantities acc(E), chi(E) and the fidelity B(rho_0,rho_1).

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arXiv2007

Quantum Complexity of Testing Group Commutativity

We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in O (k^{2/3}). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Omega(k^{2/3}), we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.

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arXiv2006

Invertible Quantum Operations and Perfect Encryption of Quantum States

In this note, we characterize the form of an invertible quantum operation, i.e., a completely positive trace preserving linear transformation (a CPTP map) whose inverse is also a CPTP map. The precise form of such maps becomes important in contexts such as self-testing and encryption. We show that these maps correspond to applying a unitary transformation to the state along with an ancilla initialized to a fixed state, which may be mixed. The characterization of invertible quantum operations implies that one-way schemes for encrypting quantum states using a classical key may be slightly more general than the ``private quantum channels'' studied by Ambainis, Mosca, Tapp and de Wolf (FOCS 2000). Nonetheless, we show that their results, most notably a lower bound of 2n bits of key to encrypt n quantum bits, extend in a straightforward manner to the general case.

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arXiv2006

Approximate Randomization of Quantum States With Fewer Bits of Key

Randomization of quantum states is the quantum analogue of the classical one-time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d/epsilon^2) Pauli operators to map any d-dimensional state to a state that is within trace distance epsilon of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter (2004), and Ambainis and Smith (2004). Then, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d-dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small epsilon.

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