Towards Quantum Logarithm Number Systems

Quantum computers, which process qubits, offer the promise of spectacular performance improvement over ordinary computers that deal only with classical bits, but there are obstacles to this vision. First, current quantum technology only allows a small number of qubits, and these are susceptible to noise. Second, quantum algorithms must be reversible, which often requires ancillary data that consume precious qubits. Third, interesting algorithms amenable to quantum implementation, such as chemistry simulation, require representing real numbers. Although quantum integer arithmetic has been studied extensively, the few works on quantum floating point demand more ancillary qubits than input data making floating point impractical for current quantum hardware. This paper suggests an alternative to floating point, known as the Logarithmic Number System (LNS), which has proven effective for approximate arithmetic with classical hardware. Reversible LNS multiplication and division are easy and exact with one ancillary qubit. Here we explore the quantum cost of difficult LNS operations (addition and subtraction). LNS offers implementation tradeoffs between accuracy and qubit cost that suggest highly-approximate LNS will be practical on quantum hardware sooner than when quantum technology has improved enough for floating-point to be practical.