Pranab Sen

Hidden translation and orbit coset in quantum computing

We give efficient quantum algorithms for the problems of <sc>Hidden Translation</sc> and <sc>Hidden Subgroup</sc> in a large class of non-abelian groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently <sc>Hidden Translation</sc> in Z <inf>p</inf>ⁿ, whenever <i>p</i> is a fixed prime. For the induction step, we introduce the problem <sc>Orbit Coset</sc> generalizing both <sc>Hidden Translation</sc> and <sc>Hidden Subgroup</sc>, and prove a powerful self-reducibility result: <sc>Orbit Coset</sc> in a finite group <i>G</i> is reducible to <sc>Orbit Coset</sc> in <i>G/N</i> and subgroups of <i>N</i>, for any solvable normal subgroup <i>N</i> of <i>G</i>.

Limitations of quantum coset states for graph isomorphism

It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [30] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n,Fpm) and Gn where G is finite and satisfies a suitable property.

Prior entanglement, message compression and privacy in quantum communication

Consider a two-party quantum communication protocol for computing some function f : {0, 1}/sup n/ /spl times/ {0, 1}/sup n/ /spl rarr/ Z. We show that the first message of P can be compressed to 0(k) classical bits using prior entanglement if it carries at most k bits of information about the senders input. This implies a general direct sum result for one-round and simultaneous quantum protocols. It also implies a new round elimination lemma in quantum communication, which allows us to extend recent classical lower bounds on the cell probe complexity of some data structure problems, e.g. approximate nearest neighbor searching on the Hamming cube {0, 1}/sup n/, to the quantum setting. We then show an optimal tradeoff between the privacy losses of Alice and Bob in computing f in terms of the one-round quantum communication complexity of f with prior entanglement. This tradeoff is independent of the number of rounds of communication. The above message compression and privacy tradeoff results use a lot of qubits of prior entanglement, leading one to wonder how much prior entanglement is really required by a quantum protocol. We show that Newmans [1991] technique of reducing the number of public coins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of black-box reduction of prior entanglement that subsumes Newmans technique. Intuitively, a black-box reduction does not change the unitary transforms of Alice and Bob; it only decreases the amount of entanglement of the prior entangled state. We prove that such a black-box reduction is impossible for quantum protocols by exhibiting a particular one-round quantum protocol for the equality function where the black-box technique fails to reduce the amount of prior entanglement by more than a constant factor.

A property of quantum relative entropy with an application to privacy in quantum communication

We prove the following information-theoretic property about quantum states. <i>Substate theorem:</i> Let ρ and σ be quantum states in the same Hilbert space with relative entropy <i>S</i>(ρ ∥ σ) ≔ Tr ρ (log ρ− log σ) = <i>c</i>. Then for all ε > 0, there is a state ρ′ such that the trace distance ∥ρ′ − ρ∥_tr ≔ Tr &sqrt;(ρ′ − ρ)² ≤ ε, and ρ′/2^{<i>O</i>(<i>c</i>/ε²)} ≤ σ. It states that if the relative entropy of ρ and σ is small, then there is a state ρ′ close to ρ, i.e. with small trace distance ∥ρ′ − ρ∥_tr, that when scaled down by a factor 2^{<i>O</i>(<i>c</i>)} ‘sits inside’, or becomes a ‘substate’ of, σ. This result has several applications in quantum communication complexity and cryptography. Using the substate theorem, we derive a privacy trade-off for the <i>set membership problem</i> in the two-party quantum communication model. Here Alice is given a subset <i>A</i> &subse; [<i>n</i>], Bob an input <i>i</i> ∈ [<i>n</i>], and they need to determine if <i>i</i> ∈ <i>A</i>. <i>Privacy trade-off for set membership:</i> In any two-party quantum communication protocol for the set membership problem, if Bob reveals only <i>k</i> bits of information about his input, then Alice must reveal at least <i>n</i>/2^O(<i>k</i>) bits of information about her input. We also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem.

Classical Communication Over a Quantum Interference Channel

Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of “very strong” and “strong” interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of “very strong” and “strong” interference, by exploiting Winters successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple-access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates.

On the power of random bases in fourier sampling: hidden subgroup problem in the heisenberg group

The hidden subgroup problem (HSP) offers a unified framework to study problems of group-theoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an efficient solution in the abelian case, not much is known for general non-abelian groups. Recently, some authors raised the question as to whether post-processing the Fourier spectrum by measuring in a random orthonormal basis helps for solving the HSP. Several negative results on the shortcomings of this random strong method are known. In this paper however, we show that the random strong method can be quite powerful under certain conditions on the group G. We define a parameter r(G) and show that O((log |G| / r(G))2) iterations of the random strong method give enough classical information to solve the HSP. We illustrate the power of the random strong method via a concrete example of the HSP over finite Heisenberg groups. We show that r(G) = Ω(1) for these groups; hence the HSP can be solved using polynomially many random strong Fourier samplings followed by a possibly exponential classical post-processing without further queries. The quantum part of our algorithm consists of a polynomial computation followed by measuring in a random orthonormal basis. As an interesting by-product of our work, we get an algorithm for solving the state identification problem for a set of nearly orthogonal pure quantum states.

Quantum Testers for Hidden Group Properties

We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the common coset range property of two functions having identical ranges within each coset of some normal subgroup. Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers. This is achieved by introducing a technique refining the majority correction process widely used for proving robustness of algebraic properties. The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.

Making Classical Honest Verifier Zero Knowledge Protocols Secure against Quantum Attacks

We show that any problem that has a classical zero-knowledge protocol against the honest verifier also has, under a reasonable condition, a classical zero-knowledge protocol which is secure against all classical and quantum polynomial time verifiers, even cheating ones. Here we refer to the generalized notion of zero-knowledge with classical and quantum auxiliary inputs respectively. Our condition on the original protocol is that, for positive instances of the problem, the simulated message transcript should be quantum computationally indistinguishable from the actual message transcript. This is a natural strengthening of the notion of honest verifier computational zero-knowledge, and includes in particular, the complexity class of honest verifier statistical zero-knowledge. Our result answers an open question of Watrous [Wat06], and generalizes classical results by Goldreich, Sahai and Vadhan [GSV98], and Vadhan [Vad06] who showed that honest verifier statistical, respectively computational, zero knowledge is equal to general statistical, respectively computational, zero knowledge.

One-Shot Marton Inner Bound for Classical-Quantum Broadcast Channel

We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of hypothesis testing and one-shot max divergences. These results give a full justification of the claims of Savov and Wilde in the classical-quantum asymptotic iid setting; the techniques also yield similar bounds in the information spectrum setting. We obtain these results using a different analysis of the random codebook argument; our method yields a classical one-shot Marton bound with a common message and a classical one-shot mutual covering lemma based on rejection sampling.

From Low-Distortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking

The existence of quantum uncertainty relations is the essential reason that some classically unrealizable cryptographic primitives become realizable when quantum communication is allowed. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking [DiVincenzo et al. 2004]. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be “locked”. Furthermore, knowing the key, it is possible to recover, that is “unlock”, the message. In this article, we make the following contributions by exploiting a connection between uncertainty relations and low-distortion embeddings of Euclidean spaces into slightly larger spaces endowed with the ℓ1 norm. We introduce the notion of a metric uncertainty relation and connect it to low-distortion embeddings of ℓ2 into ℓ1. A metric uncertainty relation also implies an entropic uncertainty relation. We prove that random bases satisfy uncertainty relations with a stronger definition and better parameters than previously known. Our proof is also considerably simpler than earlier proofs. We then apply this result to show the existence of locking schemes with key size independent of the message length. Moreover, we give efficient constructions of bases satisfying metric uncertainty relations. The bases defining these metric uncertainty relations are computable by quantum circuits of almost linear size. This leads to the first explicit construction of a strong information locking scheme. These constructions are obtained by adapting an explicit norm embedding due to Indyk [2007] and an extractor construction of Guruswami et al. [2009]. We apply our metric uncertainty relations to exhibit communication protocols that perform equality testing of n-qubit states. We prove that this task can be performed by a single message protocol using O(log2 n) qubits and n bits of communication, where the computation of the sender is efficient.