Reversible Pebbling Game for Quantum Memory Management

Abstract

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.

Improved Quantum Circuits for Elliptic Curve Discrete Logarithms

“…Several factors of reduction in T -count and T -depth are possible, while the increase in the number of qubits is significant. However, such a design point can be of high interest, in particular when combined with automatic quantum memory strategies, e.g., pebbling [20], that can find intermediate trade-off points that lie in between the manual and automatic construction.…”

The Role of Multiplicative Complexity in Compiling Low $T$-count Oracle Circuits

“…They must be restored to |0 as only input and output states must be accessible at the end of the computation. Different automatic clean-up strategies are available in the literature [13], [18], exploring the trade-off between ancillae and operations.…”

“…. , x n+r Output: Quantum circuit for U f 1 function compute is CNOT(x, t 2 ) for all x ∈ L 2 \ {t 2 }; 12 if p then NOT(t 1 ); 13 if q then NOT(t 2 ); 14 TOFFOLI(t 1 , t 2 , x i ); 15 if p then NOT(t 2 ); 16 if q then NOT(t 1 ); 17 CNOT(x, t 2 ) for all x ∈ L 2 \ {t 2 }; 18 CNOT(x, t 1 ) for all x ∈ L 1 \ {t 1 }; 19 end 20 end 21 compute; 22 CNOT(x n+r , y); 23 if p = 0 then NOT(y); 24 compute † ;…”

“…A SAT-based reversible pebbling strategy allows us to reduce the number of qubits by trading off T -count. Compared to existing reversible pebbling strategies (see, e.g., [21], [18]), we enforce that all XOR operations are performed in-place.…”

“…A weighted pebble game in which computations of AND gates are more expensive than computations of XOR gates can help to find solutions with fewer T gates, however, it comes with an overhead in solving time. Besides the (red) data point obtained by the heuristic compilation algorithm, there are 3 more Pareto optimal solutions, namely (18,296), (20,280), and (28,272). While all solutions were found within a few seconds, we noticed that the SAT-based pebbling strategy does not yet scale well to larger benchmarks such as the EPFL benchmarks.…”