13 research outputs found
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
Quantum-secure message authentication via blind-unforgeability
Formulating and designing unforgeable authentication of classical messages in
the presence of quantum adversaries has been a challenge, as the familiar
classical notions of unforgeability do not directly translate into meaningful
notions in the quantum setting. A particular difficulty is how to fairly
capture the notion of "predicting an unqueried value" when the adversary can
query in quantum superposition. In this work, we uncover serious shortcomings
in existing approaches, and propose a new definition. We then support its
viability by a number of constructions and characterizations. Specifically, we
demonstrate a function which is secure according to the existing definition by
Boneh and Zhandry, but is clearly vulnerable to a quantum forgery attack,
whereby a query supported only on inputs that start with 0 divulges the value
of the function on an input that starts with 1. We then propose a new
definition, which we call "blind-unforgeability" (or BU.) This notion matches
"intuitive unpredictability" in all examples studied thus far. It defines a
function to be predictable if there exists an adversary which can use
"partially blinded" oracle access to predict values in the blinded region. Our
definition (BU) coincides with standard unpredictability (EUF-CMA) in the
classical-query setting. We show that quantum-secure pseudorandom functions are
BU-secure MACs. In addition, we show that BU satisfies a composition property
(Hash-and-MAC) using "Bernoulli-preserving" hash functions, a new notion which
may be of independent interest. Finally, we show that BU is amenable to
security reductions by giving a precise bound on the extent to which quantum
algorithms can deviate from their usual behavior due to the blinding in the BU
security experiment.Comment: 23+9 pages, v3: published version, with one theorem statement in the
summary of results correcte
SQUARE: Strategic Quantum Ancilla Reuse for Modular Quantum Programs via Cost-Effective Uncomputation
Compiling high-level quantum programs to machines that are size constrained
(i.e. limited number of quantum bits) and time constrained (i.e. limited number
of quantum operations) is challenging. In this paper, we present SQUARE
(Strategic QUantum Ancilla REuse), a compilation infrastructure that tackles
allocation and reclamation of scratch qubits (called ancilla) in modular
quantum programs. At its core, SQUARE strategically performs uncomputation to
create opportunities for qubit reuse.
Current Noisy Intermediate-Scale Quantum (NISQ) computers and forward-looking
Fault-Tolerant (FT) quantum computers have fundamentally different constraints
such as data locality, instruction parallelism, and communication overhead. Our
heuristic-based ancilla-reuse algorithm balances these considerations and fits
computations into resource-constrained NISQ or FT quantum machines, throttling
parallelism when necessary. To precisely capture the workload of a program, we
propose an improved metric, the "active quantum volume," and use this metric to
evaluate the effectiveness of our algorithm. Our results show that SQUARE
improves the average success rate of NISQ applications by 1.47X. Surprisingly,
the additional gates for uncomputation create ancilla with better locality, and
result in substantially fewer swap gates and less gate noise overall. SQUARE
also achieves an average reduction of 1.5X (and up to 9.6X) in active quantum
volume for FT machines.Comment: 14 pages, 10 figure
Gate-based Quantum Computing for Protein Design
Protein design is a technique to engineer proteins by modifying their
sequence to obtain novel functionalities. In this method, amino acids in the
sequence are permutated to find the low energy states satisfying the
configuration. However, exploring all possible combinations of amino acids is
generally impossible to achieve on conventional computers due to the
exponential growth of possibilities with the number of designable sites. Thus,
sampling methods are currently used as a conventional approach to address the
protein design problems. Recently, quantum computation methods have shown the
potential to solve similar types of problems. In the present work, we use the
general idea of Grover's algorithm, a pure quantum computation method, to
design circuits at the gate-based level and address the protein design problem.
In our quantum algorithms, we use custom pair-wise energy tables consisting of
eight different amino acids. Also, the distance reciprocals between designable
sites are included in calculating energies in the circuits. Due to the noisy
state of current quantum computers, we mainly use quantum computer simulators
for this study. However, a very simple version of our circuits is implemented
on real quantum devices to examine their capabilities to run these algorithms.
Our results show that using iterations, the circuits
find the correct results among all possibilities, providing the expected
quadratic speed up of Grover's algorithm over classical methods
Quantum-secure message authentication via blind-unforgeability
Formulating and designing unforgeable authentication of classical messages in the presence of quantum adversaries has been a challenge, as the familiar classical notions of unforgeability do not directly translate into meaningful notions in the quantum setting. A particular difficulty is how to fairly capture the notion of "predicting an unqueried value" when the adversary can query in quantum superposition. In this work, we uncover serious shortcomings in existing approaches, and propose a new definition. We then support its viability by a number of constructions and characterizations. Specifically, we demonstrate a function wh
Tower: Data Structures in Quantum Superposition
Emerging quantum algorithms for problems such as element distinctness, subset
sum, and closest pair demonstrate computational advantages by relying on
abstract data structures. Practically realizing such an algorithm as a program
for a quantum computer requires an efficient implementation of the data
structure whose operations correspond to unitary operators that manipulate
quantum superpositions of data.
To correctly operate in superposition, an implementation must satisfy three
properties -- reversibility, history independence, and bounded-time execution.
Standard implementations, such as the representation of an abstract set as a
hash table, fail these properties, calling for tools to develop specialized
implementations.
In this work, we present Core Tower, the first language for quantum
programming with random-access memory. Core Tower enables the developer to
implement data structures as pointer-based, linked data. It features a
reversible semantics enabling every valid program to be translated to a unitary
quantum circuit.
We present Boson, the first memory allocator that supports reversible,
history-independent, and constant-time dynamic memory allocation in quantum
superposition. We also present Tower, a language for quantum programming with
recursively defined data structures. Tower features a type system that bounds
all recursion using classical parameters as is necessary for a program to
execute on a quantum computer.
Using Tower, we implement Ground, the first quantum library of data
structures, including lists, stacks, queues, strings, and sets. We provide the
first executable implementation of sets that satisfies all three mandated
properties of reversibility, history independence, and bounded-time execution.Comment: 30 pages, 22 figures. [v2] add discussion of concurrent work in Sec
1.4 and add acknowledgements section. [v3] camera-ready version, incorporates
revisions following conference revie