Efficient Distributed Quantum Computing

Efficient Distributed Quantum Computing Steve Brierley Heilbronn Institute, Dept. Mathematics, University of Bristol October 2013 Work with Robert Beals, Oliver Gray, Aram Harrow, Samuel Kutin, Noah Linden, Dan Shepherd & Mark Stather

Summary Two models of Quantum computation: Distributed Quantum Computing Quantum Parallel RAM

Summary Two models of Quantum computation: Distributed Quantum Computing Quantum Parallel RAM Result: Theses two models are efficiently related to the standard quantum circuit model Method: We use techniques from (classical) parallel computing - sorting networks. Applications: Build a quantum computer from a network of small parts Q PRAM is a new tool in quantum algorithm design

Distributed quantum computing Quantum circuits Up to N/2 two-qubit gates on any disjoint pair of qubits Not physical because any two qubits can interact

Distributed quantum computing Quantum circuits Up to N/2 two-qubit gates on any disjoint pair of qubits Not physical because any two qubits can interact Distributed quantum computing (DQC) N small processors interact in a fixed low-degree topology

Distributed quantum computing

Circuits on a DQC Suppose we want to implement the circuit C = U 1 U 2 U 3 on a 1-D nearest neighbour graph The naive approach: use SWAP gates to move gates one at a time If there are N qubits the cost is O(N 2 ) per timestep

Overview of our approach Replace the circuit with ( ) ( C P 1 U1 L P1 1 P 2 U2 L P2 1 ) ( P 3 U L 3 P 1 3 ) where U L k are local unitaries We can combine some of the permutations C P 1 U L 1 P 2 U L 2 P 3 U L 3 P 4

Overview of our approach Replace the circuit with ( ) ( C P 1 U1 L P1 1 P 2 U2 L P2 1 ) ( P 3 U L 3 P 1 3 ) where U L k are local unitaries We can combine some of the permutations C P 1 U1 L P 2 U2 L P 3 U3 L P 4 The key idea is to use a sorting network to implement P k The algorithm is universal The cost depends on the graph but is close to optimal For a 1D nearest neighbour graph the overhead is O(N)

Sorting networks A fixed network of binary comparators: if x < y then swap x, y Insertion sort Bitonic Sort

Example

Example Suppose we want to implement the circuit C = U 1 U 2 U 3 on a 1-D nearest neighbour graph Our approach yields

Emulating circuits on a fixed architecture Given an architecture constrained by G, what is the cost of emulating a highly parallel circuit? Theorem: 1) Any circuit can be emulated on a restricted architecture with a overhead depth factor of D G (the cost of a sorting network). 2) If you can do better, you have a better sorting algorithm!

Interesting architectures The cost depends on the graph... Graph Degree Routing Cost 1D n.n. 2 Naive approach O(N 2 ) 1D n.n. 2 Insertion sort O(N) 2D n.n. 4 Insertion sort O( N) Hypercube log N Bitonic sort O(log 2 N) Cyclic butterfly 4 Benes + insertion O(log N) Complete graph N n/a 1

Lull

QPRAM on a distributed quantum computer QPRAM = Circuit model + Parallel access to quantum RAM

QPRAM on a distributed quantum computer QPRAM = Circuit model + Parallel access to quantum RAM Key primitive: The global state of the computer has registers j 1,..., j N, x 1,..., x N and y 1,..., y N Locally, processor i controls j i, x i, y i. Processor i wants to query the memory at processor j i. Want to replace y i with y i x ji according to the quantum state j 1,..., j N

Algorithm for parallel memory look-ups Idea: Make the sorting network reversible Each node requires S D G T T log log N Then the same network works for all inputs We can input a superposition of destinations

Algorithm for parallel memory look-ups Each processor submits question (j i, Q, y i, 0) and answer (i, A, 0, x i ) packets

Algorithm for parallel memory look-ups Each processor submits question (j i, Q, y i, 0) and answer (i, A, 0, x i ) packets Sort the packets (with a sorting network) based on first two indices (Q < A) The sequence is now... (j, Q, y, 0)(j, Q, y, 0)... (j, Q, y, 0)(j, A, 0, x j )... Broadcast the answer x j using local CNOTs in O(log N) time CNOT each x j value to the y register

Distributed quantum memory Theorem: 1) In the circuit model, the cost of parallel memory access is O(log N log log N) 2) To access even a single piece of quantum data costs Ω(log N)

Distributed quantum memory Theorem: 1) In the circuit model, the cost of parallel memory access is O(log N log log N) 2) To access even a single piece of quantum data costs Ω(log N) Applications: MultiGrover algorithm Element Distinctness problem

Application: MultiGrover Multiple processors can Grover search the same database held in quantum memory! The first thing each processor does is form x i x i D

Application: MultiGrover Multiple processors can Grover search the same database held in quantum memory! The first thing each processor does is form x i x i D If D requires N log N qubits to store, MultiGrover finds N solutions in the same time as Grover finds 1. i.e. we have recovered the situation when the database is simple to represent.

Application: Element Distinctness Best Oracle complexity is T = O(N 2/3 ) but this requires S = O(N 2/3 ). When the function is easy to compute but hard to invert, ST 2 = O(N 2 ) Grover-Rudolph complain that we can achieve this with non-communicating parallel Grover searches

Summary Two models of Quantum computation: Distributed Quantum Computing Quantum Parallel RAM Result: Using sorting networks, the two models are efficiently related to the standard quantum circuit model Applications: Build a quantum computer from a network of small parts Q PRAM is a new tool in quantum algorithm design 1D n.n graph : Hirata et al. QIC 11, 142 (2011) Any graph & QPRAM: Beals et al. Proc. R. Soc. A (arxiv:1207.2307) Cyclic Butterfly : work in progress