30 research outputs found
Reversible Pebbling Game for Quantum Memory Management
Quantum memory management is becoming a pressing problem, especially given
the recent research effort to develop new and more complex quantum algorithms.
The only existing automatic method for quantum states clean-up relies on the
availability of many extra resources. In this work, we propose an automatic
tool for quantum memory management. We show how this problem exactly matches
the reversible pebbling game. Based on that, we develop a SAT-based algorithm
that returns a valid clean-up strategy, taking the limitations of the quantum
hardware into account. The developed tool empowers the designer with the
flexibility required to explore the trade-off between memory resources and
number of operations. We present three show-cases to prove the validity of our
approach. First, we apply the algorithm to straight-line programs, widely used
in cryptographic applications. Second, we perform a comparison with the
existing approach, showing an average improvement of 52.77%. Finally, we show
the advantage of using the tool when synthesizing a quantum circuit on a
constrained near-term quantum device.Comment: In Proc. Design Automation and Test in Europe (DATE 2019
Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t+1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system
Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling
We establish an exactly tight relation between reversible pebblings of graphs
and Nullstellensatz refutations of pebbling formulas, showing that a graph
can be reversibly pebbled in time and space if and only if there is a
Nullstellensatz refutation of the pebbling formula over in size and
degree (independently of the field in which the Nullstellensatz refutation
is made). We use this correspondence to prove a number of strong size-degree
trade-offs for Nullstellensatz, which to the best of our knowledge are the
first such results for this proof system
The Parallel Reversible Pebbling Game: Analyzing the Post-Quantum Security of iMHFs
The classical (parallel) black pebbling game is a useful abstraction which allows us to analyze the resources (space, space-time, cumulative space) necessary to evaluate a function with a static data-dependency graph . Of particular interest in the field of cryptography are data-independent memory-hard functions which are defined by a directed acyclic graph (DAG) and a cryptographic hash function . The pebbling complexity of the graph characterizes the amortized cost of evaluating multiple times as well as the total cost to run a brute-force preimage attack over a fixed domain , i.e., given find such that . While a classical attacker will need to evaluate the function at least times a quantum attacker running Grover\u27s algorithm only requires blackbox calls to a quantum circuit evaluating the function . Thus, to analyze the cost of a quantum attack it is crucial to understand the space-time cost (equivalently width times depth) of the quantum circuit . We first observe that a legal black pebbling strategy for the graph does not necessarily imply the existence of a quantum circuit with comparable complexity --- in contrast to the classical setting where any efficient pebbling strategy for corresponds to an algorithm with comparable complexity for evaluating . Motivated by this observation we introduce a new parallel reversible pebbling game which captures additional restrictions imposed by the No-Deletion Theorem in Quantum Computing. We apply our new reversible pebbling game to analyze the reversible space-time complexity of several important graphs: Line Graphs, Argon2i-A, Argon2i-B, and DRSample. Specifically, (1) we show that a line graph of size has reversible space-time complexity at most . (2) We show that any -reducible DAG has reversible space-time complexity at most . In particular, this implies that the reversible space-time complexity of Argon2i-A and Argon2i-B are at most and , respectively. (3) We show that the reversible space-time complexity of DRSample is at most . We also study the cumulative pebbling cost of reversible pebblings extending a (non-reversible) pebbling attack of Alwen and Blocki on depth-reducible graphs
The Compilation of Reversible Circuits and a New Optimization Game
The focus of this thesis is reversible circuit compilation.
We will explore the use of pebble games for circuit analysis. The usefulness of this
technique is demonstrated by finding a new space bound for the Karatsuba algorithm and
more generally for any similar algorithm based on recurrence relations. A new pebble game
based on the reversible pebble game which better captures the use of in-place operations is
also presented. We also construct circuit to compute trigonometric functions based on the
CORDIC algorithm and analyze it using this game
Improved quantum circuits for elliptic curve discrete logarithms
We present improved quantum circuits for elliptic curve scalar
multiplication, the most costly component in Shor's algorithm to compute
discrete logarithms in elliptic curve groups. We optimize low-level components
such as reversible integer and modular arithmetic through windowing techniques
and more adaptive placement of uncomputing steps, and improve over previous
quantum circuits for modular inversion by reformulating the binary Euclidean
algorithm. Overall, we obtain an affine Weierstrass point addition circuit that
has lower depth and uses fewer gates than previous circuits. While previous
work mostly focuses on minimizing the total number of qubits, we present
various trade-offs between different cost metrics including the number of
qubits, circuit depth and -gate count. Finally, we provide a full
implementation of point addition in the Q# quantum programming language that
allows unit tests and automatic quantum resource estimation for all components.Comment: 22 pages, to appear in: Int'l Conf. on Post-Quantum Cryptography
(PQCrypto 2020
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
The role of multiplicative complexity in compiling Low T-count Oracle circuits
We present a constructive method to create quantum circuits that implement oracles |x〉|y〉|0〉 k →|x〉|y⊕f(x)〉|0〉 k for n-variable Boolean functions f with low T-count. In our method f is given as a 2-regular Boolean logic network over the gate basis {∧, ⊕, 1}. Our construction leads to circuits with a T-count that is at most four times the number of AND nodes in the network. In addition, we propose a SAT-based method that allows us to trade qubits for T gates, and explore the space/complexity trade-off of quantum circuits. Our constructive method suggests a new upper bound for the number of T gates and ancilla qubits based on the multiplicative complexity c∧(f) of the oracle function f, which is the minimum number of AND gates that is required to realize f over the gate basis {∧, ⊕, 1}. There exists a quantum circuit computing f with at most 4c∧(f)T gates using k=c∧(f) ancillae. Results known for the multiplicative complexity of Boolean functions can be transferred. We verify our method by comparing it to different state-of-the-art compilers. Finally, we present our synthesis results for Boolean functions used in quantum cryptoanalysis