arXiv subjects

Scott Aaronson

Publications and source records attributed to Scott Aaronson.

At least 73 records · Page 4Linked to original sources

arXiv2014

Forrelation: A Problem that Optimally Separates Quantum from Classical Computing

We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs ~sqrt(N)/log(N) queries (improving an ~N^{1/4} lower bound of Aaronson). Conversely, we show that this 1 versus ~sqrt(N) separation is optimal: indeed, any t-query quantum algorithm whatsoever can be simulated by an O(N^{1-1/2t})-query randomized algorithm. Thus, resolving an open question of Buhrman et al. from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal t versus ~N^{1-1/2t} separation for all t. As a bonus, we show that this generalization is BQP-complete. This yields what's arguably the simplest BQP-complete problem yet known, and gives a second sense in which Forrelation "captures the maximum power of quantum computation."

quant-ph↗

arXiv2014

Bounding the seed length of Miller and Shi's unbounded randomness expansion protocol

Recent randomness expansion protocols have been proposed which are able to generate an unbounded amount of randomness from a finite amount of truly random initial seed. One such protocol, given by Miller and Shi, uses a pair of non-signaling untrusted quantum mechanical devices. These play XOR games with inputs given by the user in order to generate an output. Here we present an analysis of the required seed size, giving explicit upper bounds for the number of initial random bits needed to jump-start the protocol. The bits output from such a protocol are $\varepsilon$-close to uniform even against quantum adversaries. Our analysis yields that for a statistical distance of $\varepsilon=10^{-1}$ and $\varepsilon=10^{-6}$ from uniformity, the number of required bits is smaller than 225,000 and 715,000, respectively; in general it grows as $O(\log\frac{1}{\varepsilon})$.

quant-ph↗

arXiv2014

Near invariance of the hypercube

We give an almost-complete description of orthogonal matrices $M$ of order $n$ that "rotate a non-negligible fraction of the Boolean hypercube $C_n=\{-1,1\}^n$ onto itself," in the sense that $$P_{x\in C_n}(Mx\in C_n) \ge n^{-C},\mbox{ for some positive constant } C,$$ where $x$ is sampled uniformly over $C_n$. In particular, we show that such matrices $M$ must be very close to products of permutation and reflection matrices. This result is a step toward characterizing those orthogonal and unitary matrices with large permanents, a question with applications to linear-optical quantum computing.

math.CO↗

arXiv2014

Quantum POMDPs

We present quantum observable Markov decision processes (QOMDPs), the quantum analogues of partially observable Markov decision processes (POMDPs). In a QOMDP, an agent's state is represented as a quantum state and the agent can choose a superoperator to apply. This is similar to the POMDP belief state, which is a probability distribution over world states and evolves via a stochastic matrix. We show that the existence of a policy of at least a certain value has the same complexity for QOMDPs and POMDPs in the polynomial and infinite horizon cases. However, we also prove that the existence of a policy that can reach a goal state is decidable for goal POMDPs and undecidable for goal QOMDPs.

cs.AI↗

arXiv2014

Generation of Universal Linear Optics by Any Beamsplitter

In 1994, Reck et al. showed how to realize any unitary transformation on a single photon using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of unitary transformations (or orthogonal transformations, in the real case) on the single-photon subspace with m>=3 modes. (We prove the same result for any two-mode real optical gate, and for any two-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics. Theoretically, it means that one cannot produce "intermediate" models of linear optical computation (analogous to the Clifford group for qubits) by restricting the allowed beamsplitters and phaseshifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on three or more modes.

quant-ph↗

arXiv2014

Quantifying the Rise and Fall of Complexity in Closed Systems: The Coffee Automaton

In contrast to entropy, which increases monotonically, the "complexity" or "interestingness" of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex structures at the Big Bang and will also lack them after black holes evaporate and particles are dispersed. This paper makes an initial attempt to quantify this pattern. As a model system, we use a simple, two-dimensional cellular automaton that simulates the mixing of two liquids ("coffee" and "cream"). A plausible complexity measure is then the Kolmogorov complexity of a coarse-grained approximation of the automaton's state, which we dub the "apparent complexity." We study this complexity measure, and show analytically that it never becomes large when the liquid particles are non-interacting. By contrast, when the particles do interact, we give numerical evidence that the complexity reaches a maximum comparable to the "coffee cup's" horizontal dimension. We raise the problem of proving this behavior analytically.

cond-mat.stat-mech↗

arXiv2014

The Need for Structure in Quantum Speedups

Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate. First, we show that for any problem that is invariant under permuting inputs and outputs (like the collision or the element distinctness problems), the quantum query complexity is at least the 7th root of the classical randomized query complexity. (An earlier version of this paper gave the 9th root.) This resolves a conjecture of Watrous from 2002. Second, inspired by recent work of O'Donnell et al. (2005) and Dinur et al. (2006), we conjecture that every bounded low-degree polynomial has a "highly influential" variable. Assuming this conjecture, we show that every T-query quantum algorithm can be simulated on most inputs by a poly(T)-query classical algorithm, and that one essentially cannot hope to prove P!=BQP relative to a random oracle.

quant-ph↗

arXiv2014

AM with Multiple Merlins

We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close analogies between it and the quantum complexity class QMA(k), but the AM(k) model is also natural in its own right. We illustrate the power of multiple Merlins by giving an AM(2) protocol for 3SAT, in which the Merlins' challenges and responses consist of only n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms nearly matching this lower bound are known, but their running times had never been previously explained. Brandao and Harrow have also recently used our 3SAT protocol to show quasipolynomial hardness for approximating the values of certain entangled games. In the other direction, we give a simple quasipolynomial-time approximation algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT protocol is essentially optimal. More generally, we show that multiple Merlins never provide more than a polynomial advantage over one: that is, AM(k)=AM for all k=poly(n). The key to this result is a subsampling theorem for free games, which follows from powerful results by Alon et al. and Barak et al. on subsampling dense CSPs, and which says that the value of any free game can be closely approximated by the value of a logarithmic-sized random subgame.

cs.CC↗

arXiv2013

Weak Parity

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in the quantum case for any eps>0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.

cs.CC↗

arXiv2013

A Full Characterization of Quantum Advice

We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.

quant-ph↗

arXiv2013

BosonSampling Is Far From Uniform

BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. In arXiv:1306.3995, Gogolin et al. claimed that even an ideal BosonSampling device's output would be "operationally indistinguishable" from a uniform random outcome, at least "without detailed a priori knowledge"; or at any rate, that telling the two apart might itself be a hard problem. We first answer these claims---explaining why the first is based on a definition of "a priori knowledge" so strange that, were it adopted, almost no quantum algorithm could be distinguished from a pure random-number source; while the second is neither new nor a practical obstacle to interesting BosonSampling experiments. However, we then go further, and address some interesting research questions inspired by Gogolin et al.'s mistaken arguments. We prove that, with high probability over a Haar-random matrix A, the BosonSampling distribution induced by A is far from the uniform distribution in total variation distance. More surprisingly, and directly counter to Gogolin et al., we give an efficient algorithm that distinguishes these two distributions with constant bias. Finally, we offer three "bonus" results about BosonSampling. First, we report an observation of Fernando Brandao: that one can efficiently sample a distribution that has large entropy and that's indistinguishable from a BosonSampling distribution by any circuit of fixed polynomial size. Second, we show that BosonSampling distributions can be efficiently distinguished from uniform even with photon losses and for general initial states. Third, we offer the simplest known proof that FermionSampling is solvable in classical polynomial time, and we reuse techniques from our BosonSampling analysis to characterize random FermionSampling distributions.

quant-ph↗

arXiv2013

Psi-Epistemic Theories: The Role of Symmetry

Formalizing an old desire of Einstein, "psi-epistemic theories" try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called "ontic states." Regardless of one's philosophical views about such theories, the question arises of whether one can cleanly rule them out, by proving no-go theorems analogous to the Bell Inequality. In the 1960s, Kochen and Specker (who first studied these theories) constructed an elegant psi-epistemic theory for Hilbert space dimension d=2, but also showed that any deterministic psi-epistemic theory must be "measurement contextual" in dimensions 3 and higher. Last year, the topic attracted renewed attention, when Pusey, Barrett, and Rudolph (PBR) showed that any psi-epistemic theory must "behave badly under tensor product." In this paper, we prove that even without the Kochen-Specker or PBR assumptions, there are no psi-epistemic theories in dimensions d>=3 that satisfy two reasonable conditions: (1) symmetry under unitary transformations, and (2) "maximum nontriviality" (meaning that the probability distributions corresponding to any two non-orthogonal states overlap). This no-go theorem holds if the ontic space is either the set of quantum states or the set of unitaries. The proof of this result, in the general case, uses some measure theory and differential geometry. On the other hand, we also show the surprising result that without the symmetry restriction, one can construct maximally-nontrivial psi-epistemic theories in every finite dimension d.

quant-ph↗

arXiv2013

The Ghost in the Quantum Turing Machine

In honor of Alan Turing's hundredth birthday, I unwisely set out some thoughts about one of Turing's obsessions throughout his life, the question of physics and free will. I focus relatively narrowly on a notion that I call "Knightian freedom": a certain kind of in-principle physical unpredictability that goes beyond probabilistic unpredictability. Other, more metaphysical aspects of free will I regard as possibly outside the scope of science. I examine a viewpoint, suggested independently by Carl Hoefer, Cristi Stoica, and even Turing himself, that tries to find scope for "freedom" in the universe's boundary conditions rather than in the dynamical laws. Taking this viewpoint seriously leads to many interesting conceptual problems. I investigate how far one can go toward solving those problems, and along the way, encounter (among other things) the No-Cloning Theorem, the measurement problem, decoherence, chaos, the arrow of time, the holographic principle, Newcomb's paradox, Boltzmann brains, algorithmic information theory, and the Common Prior Assumption. I also compare the viewpoint explored here to the more radical speculations of Roger Penrose. The result of all this is an unusual perspective on time, quantum mechanics, and causation, of which I myself remain skeptical, but which has several appealing features. Among other things, it suggests interesting empirical questions in neuroscience, physics, and cosmology; and takes a millennia-old philosophical debate into some underexplored territory.

quant-ph↗

arXiv2013

Photonic Boson Sampling in a Tunable Circuit

Quantum computers are unnecessary for exponentially-efficient computation or simulation if the Extended Church-Turing thesis---a foundational tenet of computer science---is correct. The thesis would be directly contradicted by a physical device that efficiently performs a task believed to be intractable for classical computers. Such a task is BosonSampling: obtaining a distribution of n bosons scattered by some linear-optical unitary process. Here we test the central premise of BosonSampling, experimentally verifying that the amplitudes of 3-photon scattering processes are given by the permanents of submatrices generated from a unitary describing a 6-mode integrated optical circuit. We find the protocol to be robust, working even with the unavoidable effects of photon loss, non-ideal sources, and imperfect detection. Strong evidence against the Extended Church-Turing thesis will come from scaling to large numbers of photons, which is a much simpler task than building a universal quantum computer.

quant-ph↗

arXiv2012

Generalizing and Derandomizing Gurvits's Approximation Algorithm for the Permanent

Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an n*n matrix A. The algorithm runs in O(n^2/eps^2) time, and approximates Per(A) to within eps*||A||^n additive error. A major advantage of Gurvits's algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. This makes it highly relevant to quantum optics, where the permanents of bounded-norm complex matrices play a central role. Indeed, the existence of Gurvits's algorithm is why, in their recent work on the hardness of quantum optics, Aaronson and Arkhipov (AA) had to talk about sampling problems rather than estimation problems. In this paper, we improve Gurvits's algorithm in two ways. First, using an idea from quantum optics, we generalize the algorithm so that it yields a better approximation when the matrix A has either repeated rows or repeated columns. Translating back to quantum optics, this lets us classically estimate the probability of any outcome of an AA-type experiment---even an outcome involving multiple photons "bunched" in the same mode---at least as well as that probability can be estimated by the experiment itself. (This does not, of course, let us solve the AA sampling problem.) It also yields a general upper bound on the probabilities of "bunched" outcomes, which resolves a conjecture of Gurvits and might be of independent physical interest. Second, we use eps-biased sets to derandomize Gurvits's algorithm, in the special case where the matrix A is nonnegative. More interestingly, we generalize the notion of eps-biased sets to the complex numbers, construct "complex eps-biased sets," then use those sets to derandomize even our generalization of Gurvits's algorithm to the multirow/multicolumn case (again for nonnegative A). Whether Gurvits's algorithm can be derandomized for general A remains an outstanding problem.

quant-ph↗

arXiv2012

Quantum Money from Hidden Subspaces

Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is (1) public-key, meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and (2) cryptographically secure, under a "classical" hardness assumption that has nothing to do with quantum money. Our scheme is based on hidden subspaces, encoded as the zero-sets of random multivariate polynomials. A main technical advance is to show that the "black-box" version of our scheme, where the polynomials are replaced by classical oracles, is unconditionally secure. Previously, such a result had only been known relative to a quantum oracle (and even there, the proof was never published). Even in Wiesner's original setting -- quantum money that can only be verified by the bank -- we are able to use our techniques to patch a major security hole in Wiesner's scheme. We give the first private-key quantum money scheme that allows unlimited verifications and that remains unconditionally secure, even if the counterfeiter can interact adaptively with the bank. Our money scheme is simpler than previous public-key quantum money schemes, including a knot-based scheme of Farhi et al. The verifier needs to perform only two tests, one in the standard basis and one in the Hadamard basis -- matching the original intuition for quantum money, based on the existence of complementary observables. Our security proofs use a new variant of Ambainis's quantum adversary method, and several other tools that might be of independent interest.

quant-ph↗

arXiv2011

A Counterexample to the Generalized Linial-Nisan Conjecture

In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth circuits. The original Linial-Nisan Conjecture was recently proved by Braverman; we offered a $200 prize for the generalized version. In this paper, we save ourselves $200 by showing that the GLN Conjecture is false, at least for circuits of depth 3 and higher. As a byproduct, our counterexample also implies that Pi2P is not contained in P^NP relative to a random oracle with probability 1. It has been conjectured since the 1980s that PH is infinite relative to a random oracle, but the highest levels of PH previously proved separate were NP and coNP. Finally, our counterexample implies that the famous results of Linial, Mansour, and Nisan, on the structure of AC0 functions, cannot be improved in several interesting respects.

cs.CC↗

arXiv2011

Quantum Copy-Protection and Quantum Money

Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner's scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copy-protected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a "Complexity-Theoretic No-Cloning Theorem," which generalizes both the standard No-Cloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum t-designs. Moving beyond the oracle world, we also present an explicit candidate scheme for publicly-verifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copy-protecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption.)

quant-ph↗