A quantum random access memory (QRAM) using a polynomial encoding of binary strings

A quantum random access memory (QRAM) using a polynomial encoding of binary strings

Blog Article

Abstract Quantum algorithms claim significant speedup over their classical counterparts for solving many problems.An important here aspect of many of these algorithms is the existence of a quantum oracle, which needs to be implemented efficiently in order to realize the claimed advantages in practice.A quantum random access memory (QRAM) is a promising architecture for realizing these oracles.In this paper we develop a new design for QRAM and implement it with Clifford+T circuit.

We focus on optimizing the T-count and T-depth since non-Clifford gates are the most expensive to implement fault-tolerantly in most error correction schemes.Integral to our design is a polynomial encoding of bit strings and so we refer to this design as $$ ext {QRAM}_{poly}$$.Compared to the previous state-of-the-art bucket brigade architecture for QRAM, we achieve an exponential improvement in T-depth, while reducing T-count and keeping the qubit-count same.Specifically, if N is the number of memory locations to be queried, then $$ ext {QRAM}_{poly}$$ has T-depth $$O(log log N)$$ , T-count $$O(N-log N)$$ and uses O(N) logical qubits, while the bucket brigade circuit has T-depth $$O(log N)$$ , T-count O(N) and uses O(N) qubits.

Combining two $$ ext {QRAM}_{poly}$$ we design a quantum look-up-table, $$ ext {qLUT}_{poly}$$ , that has T-depth $$O(log log N)$$ , T-count $$O(sqrt{N})$$ and qubit count $$O(sqrt{N})$$.A quantum look-up table (qLUT) or quantum read-only memory (QROM) has restricted functionality than a QRAM.For example, it cannot write into a memory location and the circuit needs to be compiled each time the contents of the memory change.The previous state-of-the-art CSWAP architecture has T-depth $$O(sqrt{N})$$ , T-count $$O(sqrt{N})$$ and qubit count $$O(sqrt{N})$$.

Thus we achieve a double exponential improvement in T-depth while keeping the T-count and qubit-count asymptotically same.Additionally, with here our polynomial encoding of bit strings, we develop a method to optimize the Toffoli-count of circuits, specially those consisting of multi-controlled-NOT gates.