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The European Physical Journal D

2N qubit “mirror states” for optimal quantum communication

S. Muralidharan, S. Karumanchi, S. Jain, R. Srikanth, P. K. Panigrahi

Published Jan 14, 2011• 9 citations
Publisher
DOI
10.1140/epjd/e2010-09653-x

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We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of n qubits (held by another), up to constant accuracy, must transmit at least Ω(2n) bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an ϵ -net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires Ω(N) classical queries for an input of size N . Second, a nonlocal task which can be solved using n Bell pairs, but for which any approximate classical solution must communicate Ω(2n) bits.
By how much must the communication complexity of a function increase if we demand that the parties not only correctly compute the function but also return all registers (other than the one containing the answer) to their initial states at the end of the communication protocol? Protocols that achieve this are referred to as clean and the associated cost as the clean communication complexity. Here we present clean protocols for calculating the inner product of two n-bit strings, showing that (in the absence of preshared entanglement) at most n+3 qubits or n+O(sqrt[n]) bits of communication are required. The quantum protocol provides inspiration for obtaining the optimal method to implement distributed cnot gates in parallel while minimizing the amount of quantum communication. For more general functions, we show that nearly all Boolean functions require close to 2n bits of classical communication to compute and close to n qubits if the parties have access to preshared entanglement. Both of these values are maximal for their respective paradigms.
We present two scalable and entanglement-free methods for estimating the collective state of an n-qubit quantum computer. The first method consists of a fixed set of five quantum circuits-regardless of the number of qubits-that avoid the use of entanglement as a measurement resource, relying instead on classical communication between selected pairs of qubits. The second method requires only 2n+1 circuits, each of which applies a single local gate to one of the n qubits during the measurement stage. Unlike traditional estimation methods, our approaches do not require any costly post-processing procedure to estimate a quantum state, enabling scalability to relatively large system sizes. We experimentally compare both methods on freely available IBM quantum processors, and observe how the state estimation varies with increasing number of qubits and shots. We further validated our results by estimating the 4-qubit entangled state of two remote ion-trap quantum processors, demonstrating that the optimized 2n+1 tomographic scheme achieves estimates consistent with standard methods while using exponentially fewer measurements.