Bernoulli Factory Research Articles

Modular quantum-to-quantum Bernoulli factory in an integrated photonic processor

Modular quantum-to-quantum Bernoulli factory in an integrated photonic processor

Termination and Universal Termination Problems for Nondeterministic Quantum Programs

Verifying quantum programs has attracted a lot of interest in recent years. In this paper, we consider the following two categories of termination problems of quantum programs with nondeterminism, namely: (1) (termination) Is an input of a program terminating with probability one under all schedulers? If not, how can a scheduler be synthesized to evidence the nontermination? (2) (universal termination) Are all inputs terminating with probability one under their respective schedulers? If yes, a further question asks whether there is a scheduler that forces all inputs to be terminating with probability one together with how to synthesize it; otherwise, how can an input be provided to refute the universal termination? For the effective verification of the first category, we over-approximate the reachable set of quantum program states by the reachable subspace, whose algebraic structure is a linear space. On the other hand, we study the set of divergent states from which the program terminates with probability zero under some scheduler. The divergent set also has an explicit algebraic structure. Exploiting these explicit algebraic structures, we address the decision problem by a necessary and sufficient condition, i. e. the disjointness of the reachable subspace and the divergent set. Furthermore, the scheduler synthesis is completed in exponential time, whose bottleneck lies in computing the divergent set, reported for the first time. For the second category, we reduce the decision problem to the existence of invariant subspace, from which the program terminates with probability zero under all schedulers. The invariant subspace is characterized by linear equations and thus can be efficiently computed. The states on that invariant subspace are evidence of the nontermination. Furthermore, the scheduler synthesis is completed by seeking a pattern of finite schedulers that forces all inputs to be terminating with positive probability. The repetition of that pattern yields the desired universal scheduler that forces all inputs to be terminating with probability one. All the problems in the second category are shown, also for the first time, to be solved in polynomial time. Finally, we demonstrate the aforementioned methods via a running example — the quantum Bernoulli factory protocol.

Polarization-encoded photonic quantum-to-quantum Bernoulli factory based on a quantum dot source.

A Bernoulli factory is a randomness manipulation routine that takes as input a Bernoulli random variable, outputting another Bernoulli variable whose bias is a function of the input bias. Recently proposed quantum-to-quantum Bernoulli factory schemes encode both input and output variables in qubit amplitudes. This primitive could be used as a subroutine for more complex quantum algorithms involving Bayesian inference and Monte Carlo methods. Here, we report an experimental implementation of a polarization-encoded photonic quantum-to-quantum Bernoulli factory. We present and test three interferometric setups implementing the basic operations of an algebraic field (inversion, multiplication, and addition), which, chained together, allow for the implementation of a generic quantum-to-quantum Bernoulli factory. These in-bulk schemes are validated using a quantum dot-based single-photon source featuring high brightness and indistinguishability, paired with a time-to-spatial demultiplexing setup to prepare input resources of up to three single-photon states.

Bernoulli Factories for Flow-Based Polytopes

Bernoulli Factories for Flow-Based Polytopes

Bernoulli factories and duality in Wright–Fisher and Allen–Cahn models of population genetics

Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright–Fisher diffusion and the Kingman coalescent, where the former describes the stochastic evolution of neutral allele frequencies in a large population forwards in time, and the latter describes the genetic ancestry of randomly sampled individuals from the population backwards in time. As well as providing a richer description than either model in isolation, duality often yields equations satisfied by quantities of interest. We employ the so-called Bernoulli factory – a celebrated tool in simulation-based computing – to derive duality relations for broad classes of genetics models. As concrete examples, we present Wright–Fisher diffusions with general drift functions, and Allen–Cahn equations with general, nonlinear forcing terms. The drift and forcing functions can be interpreted as the action of frequency-dependent selection. To our knowledge, this work is the first time a connection has been drawn between Bernoulli factories and duality in models of population genetics.

Combinatorial Bernoulli factories

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter p∈[0,1] means the algorithm does not know p, but has sample access to independent draws of a Bernoulli random variable with mean equal to p. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector x∈P for a given polytope P⊂[0,1]n, output a randomized vertex such that the expected value of the i-th coordinate is exactly equal to xi. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix [xij] and asked to sample a matching such that the probability of each edge (i,j) be present in the matching is exactly equal to xij. We show that a polytope P admits a Bernoulli factory if and and only if P is the intersection of [0,1]n with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on P. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the k-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.1 1 A preliminary conference version of this work has appeared in the proceeding of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC’21) [24]. The current version presents all the missing proofs and technical details, as well as new results, alternative proofs, and more explanations.

From the Bernoulli factory to a dice enterprise via perfect sampling of Markov chains

Given a p-coin that lands heads with unknown probability p, we wish to produce an f(p)-coin for a given function f:(0,1)→(0,1). This problem is commonly known as the Bernoulli factory and results on its solvability and complexity have been obtained in (ACM Trans. Model. Comput. Simul. 4 (1994) 213–219; Ann. Appl. Probab. 15 (2005) 93–115). Nevertheless, generic ways to design a practical Bernoulli factory for a given function f exist only in a few special cases. We present a constructive way to build an efficient Bernoulli factory when f(p) is a rational function with coefficients in R. Moreover, we extend the Bernoulli factory problem to a more general setting where we have access to an m-sided die and we wish to roll a v-sided one; that is, we consider rational functions between open probability simplices. Our construction consists of rephrasing the original problem as simulating from the stationary distribution of a certain class of Markov chains—a task that we show can be achieved using perfect simulation techniques with the original m-sided die as the only source of randomness. In the Bernoulli factory case, the number of tosses needed by the algorithm has exponential tails and its expected value can be bounded uniformly in p. En route to optimizing the algorithm we show a fact of independent interest: every finite, integer valued, random variable will eventually become log-concave after convolving with enough Bernoulli trials.

Efficient Bernoulli factory Markov chain Monte Carlo for intractable posteriors

Summary Accept-reject-based Markov chain Monte Carlo algorithms have traditionally utilized acceptance probabilities that can be explicitly written as a function of the ratio of the target density at the two contested points. This feature is rendered almost useless in Bayesian posteriors with unknown functional forms. We introduce a new family of Markov chain Monte Carlo acceptance probabilities that has the distinguishing feature of not being a function of the ratio of the target density at the two points. We present two stable Bernoulli factories that generate events within this class of acceptance probabilities. The efficiency of our methods relies on obtaining reasonable local upper or lower bounds on the target density, and we present two classes of problems where such bounds are viable: Bayesian inference for diffusions, and Markov chain Monte Carlo on constrained spaces. The resulting portkey Barker’s algorithms are exact and computationally more efficient that the current state of the art.

General quantum Bernoulli factory: framework analysis and experiments

The unremitting pursuit for quantum advantages gives rise to the discovery of a quantum-enhanced randomness processing named quantum Bernoulli factory (QBF). This quantum enhanced process can show its priority over the corresponding classical process through readily available experimental resources, thus in the near term it may be capable of accelerating the applications of classical Bernoulli factories, such as the widely used sampling algorithms. In this work, we provide the framework analysis of the QBF. We thoroughly analyze the quantum state evolution in this process, discovering the field structure of the constructible quantum states. Our framework analysis shows that naturally, the previous works can be described as specific instances of this framework. Then, as a proof of principle, we experimentally demonstrate this framework via an entangled two-photon source along with a reconfigurable photonic logic, and show the advantages of the QBF over the classical model through a classically infeasible instance. These results may stimulate the discovery of advantages of the quantum randomness processing in a wider range of tasks, as well as its potential applications.

Bernoulli Factories and Black-box Reductions in Mechanism Design

We provide a polynomial time reduction from Bayesian incentive compatible mechanism design to Bayesian algorithm design for welfare maximization problems. Unlike prior results, our reduction achieves exact incentive compatibility for problems with multi-dimensional and continuous type spaces. The key technical barrier preventing exact incentive compatibility in prior black-box reductions is that repairing violations of incentive constraints requires understanding the distribution of the mechanism’s output, which is typically #P-hard to compute. Reductions that instead estimate the output distribution by sampling inevitably suffer from sampling error, which typically precludes exact incentive compatibility. We overcome this barrier by employing and generalizing the computational model in the literature on Bernoulli Factories . In a Bernoulli factory problem, one is given a function mapping the bias of an “input coin” to that of an “output coin,” and the challenge is to efficiently simulate the output coin given only sample access to the input coin. This is the key ingredient in designing an incentive compatible mechanism for bipartite matching, which can be used to make the approximately incentive compatible reduction of Hartline et al. [18] exactly incentive compatible.

Combinatorial Bernoulli Factories: Matchings, Flows and Other Polytopes

A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter p ∈ [0, 1] means the algorithm does not know p, but has sample access to independent draws of a Bernoulli random variable with mean equal to p. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector x ∈ P for a given polytope P ⊂ [0, 1]^n, output a randomized vertex such that the expected value of the i-th coordinate is exactly equal to xi. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix [x_ij] and asked to sample a matching such that the probability of each edge (i,j) be present in the matching is exactly equal to x_ij. We show that a polytope P admits a Bernoulli factory if and and only if P is the intersection of [0, 1]^n with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on P. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the k-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.

Experimental demonstration of quantum-to-quantum Bernoulli factory

A Bernoulli factory is a model of randomness processing. Recently, the quantum generalization of a Bernoulli factory with a quantum input and classical output offers a clear advantage over classical means in simulating classical randomness. A more ``quantum'' version of a Bernoulli factory with a quantum input and quantum output has been proposed and shows applications in simulating quantum randomness using quantum resources, the so-called quantum-to-quantum Bernoulli factory, which simulates probability amplitudes rather than probability. In this paper, we report an experimental implementation of a quantum-to-quantum Bernoulli factory with linear optics. We implement three elements of a quantum-to-quantum Bernoulli factory, including inversion, multiplication, and addition, and demonstrate how to compare quantum states via a quantum-to-quantum Bernoulli factory. Our results provide a thorough understanding of the Bernoulli factory problem in the quantum world and will stimulate the quantum advantages in simulating a wider range of classically infeasible random processes.

Relational proofs for quantum programs

Relational verification of quantum programs has many potential applications in quantum and post-quantum security and other domains. We propose a relational program logic for quantum programs. The interpretation of our logic is based on a quantum analogue of probabilistic couplings. We use our logic to verify non-trivial relational properties of quantum programs, including uniformity for samples generated by the quantum Bernoulli factory, reliability of quantum teleportation against noise (bit and phase flip), security of quantum one-time pad and equivalence of quantum walks.

Quantitative robustness analysis of quantum programs

Quantum computation is a topic of significant recent interest, with practical advances coming from both research and industry. A major challenge in quantum programming is dealing with errors (quantum noise) during execution. Because quantum resources (e.g., qubits) are scarce, classical error correction techniques applied at the level of the architecture are currently cost-prohibitive. But while this reality means that quantum programs are almost certain to have errors, there as yet exists no principled means to reason about erroneous behavior. This paper attempts to fill this gap by developing a semantics for erroneous quantum while-programs, as well as a logic for reasoning about them. This logic permits proving a property we have identified, called є-robustness, which characterizes possible “distance” between an ideal program and an erroneous one. We have proved the logic sound, and showed its utility on several case studies, notably: (1) analyzing the robustness of noisy versions of the quantum Bernoulli factory (QBF) and quantum walk (QW); (2) demonstrating the (in)effectiveness of different error correction schemes on single-qubit errors; and (3) analyzing the robustness of a fault-tolerant version of QBF.

An asymptotically optimal Bernoulli factory for certain functions that can be expressed as power series

Given a sequence of independent Bernoulli variables with unknown parameter p, and a function f expressed as a power series with non-negative coefficients that sum to at most 1, an algorithm is presented that produces a Bernoulli variable with parameter f(p). In particular, the algorithm can simulate f(p)=pa, a∈(0,1). For functions with a derivative growing at least as f(p)∕p for p→0, the average number of inputs required by the algorithm is asymptotically optimal among all simulations that are fast in the sense of Nacu and Peres. A non-randomized version of the algorithm is also given. Some extensions are discussed.

Quantum-to-quantum Bernoulli factory problem

Given a coin with unknown bias $p\in [0,1]$, can we exactly simulate another coin with bias $f(p)$? The exact set of simulable functions has been well characterized 20 years ago. In this paper, we ask the quantum counterpart of this question: Given the quantum coin $|p\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle$, can we exactly simulate another quantum coin $|f(p)\rangle=\sqrt{f(p)}|0\rangle+\sqrt{1-f(p)}|1\rangle$? We give the full characterization of simulable quantum state $k_0(p)|0\rangle+k_1(p)|1\rangle$ from quantum coin $|p\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle$, and present an algorithm to transform it. Surprisingly, we show that simulable sets in the quantum-to-quantum case and classical-to-classical case have no inclusion relationship with each other.

Algorithmic analysis of termination problems for quantum programs

We introduce the notion of linear ranking super-martingale (LRSM) for quantum programs (with nondeterministic choices, namely angelic and demonic choices). Several termination theorems are established showing that the existence of the LRSMs of a quantum program implies its termination. Thus, the termination problems of quantum programs is reduced to realisability and synthesis of LRSMs. We further show that the realisability and synthesis problem of LRSMs for quantum programs can be reduced to an SDP (Semi-Definite Programming) problem, which can be settled with the existing SDP solvers. The techniques developed in this paper are used to analyse the termination of several example quantum programs, including quantum random walks and quantum Bernoulli factory for random number generation. This work is essentially a generalisation of constraint-based approach to the corresponding problems for probabilistic programs developed in the recent literature by adding two novel ideas: (1) employing the fundamental Gleason's theorem in quantum mechanics to guide the choices of templates; and (2) a generalised Farkas' lemma in terms of observables (Hermitian operators) in quantum physics.

Barker’s algorithm for Bayesian inference with intractable likelihoods

In this expository paper, we abstract and describe a simple MCMC scheme for sampling from intractable target densities. The approach has been introduced in Gonçalves, Łatuszyński and Roberts (2017a) in the specific context of jump-diffusions, and is based on the Barker’s algorithm paired with a simple Bernoulli factory type scheme, the so called 2-coin algorithm. In many settings, it is an alternative to standard Metropolis–Hastings pseudo-marginal method for simulating from intractable target densities. Although Barker’s is well known to be slightly less efficient than Metropolis–Hastings, the key advantage of our approach is that it allows to implement the “marginal Barker’s” instead of the extended state space pseudo-marginal Metropolis–Hastings, owing to the special form of the accept/reject probability. We shall illustrate our methodology in the context of Bayesian inference for discretely observed Wright–Fisher family of diffusions.

Bernoulli factories and black-box reductions in mechanism design

In this letter, we report on our work providing a polynomial time reduction from Bayesian incentive compatible mechanism design to Bayesian algorithm design for welfare maximization problems. Unlike prior results, our reduction achieves exact incentive compatibility for problems with multidimensional and continuous type spaces.

Optimal Linear Bernoulli Factories for Small Mean Problems

Suppose a coin with unknown probability p of heads can be flipped as often as desired. A Bernoulli factory for a function f is an algorithm that uses flips of the coin together with auxiliary randomness to flip a single coin with probability f(p) of heads. Applications include perfect sampling from the stationary distribution of certain regenerative processes. When f is analytic, the problem can be reduced to a Bernoulli factory of the form f(p) = Cp for constant C. Presented here is a new algorithm that for small values of Cp, requires roughly only C coin flips. From information theoretic considerations, this is also conjectured to be (to first order) the minimum number of flips needed by any such algorithm. For large values of Cp, the new algorithm can also be used to build a new Bernoulli factory that uses only 80 % of the expected coin flips of the older method. In addition, the new method also applies to the more general problem of a linear multivariate Bernoulli factory, where there are k coins, the kth coin has unknown probability pk of heads, and the goal is to simulate a coin flip with probability C1p1+⋯ + Ckpk of heads.