Quantum measurement is a fundamental cornerstone of experimental quantum computations. The main issues in current quantum measurement strategies are the high number of measurement rounds to determine a global optimal measurement output and the low success probability of finding a global optimal measurement output. Each measurement round requires preparing the quantum system and applying quantum operations and measurements with high-precision control in the physical layer. These issues result in extremely high-cost measurements with a low probability of success at the end of the measurement rounds. Here, we define a novel measurement for quantum computations called dense quantum measurement. The dense measurement strategy aims at fixing the main drawbacks of standard quantum measurements by achieving a significant reduction in the number of necessary measurement rounds and by radically improving the success probabilities of finding global optimal outputs. We provide application scenarios for quantum circuits with arbitrary unitary sequences, and prove that dense measurement theory provides an experimentally implementable solution for gate-model quantum computer architectures.
An application of the QAOA algorithm to a bounded occurrence constraint problem “Max E3LIN2” can be found in @cite_39 . In the analyzed problem, the input is a set of linear equations each of which has three boolean variables, and each equation outputs whether the sum of the variables is 0 or is 1 in a mod 2 representation. The work is aimed to demonstrate the capabilities of the QAOA algorithm in a gate-model quantum computer environment.
@cite_39
1752349928 2901444101 2297565774 1973652906
We apply our recent Quantum Approximate Optimization Algorithm to the combinatorial problem of bounded occurrence Max E3LIN2. The input is a set of linear equations each of which contains exactly three boolean variables and each equation says that the sum of the variables mod 2 is 0 or is 1. Every variable is in no more than D equations. A random string will satisfy 1 2 of the equations. We show that the level one QAOA will efficiently produce a string that satisfies @math times the number of equations. A recent classical algorithm achieved @math . We also show that in the typical case the quantum computer will output a string that satisfies @math times the number of equations. The Quantum Approximate Optimization Algorithm (QAOA) is a promising approach for programming a near-term gate-based hybrid quantum computer to find good approximate solutions of hard combinatorial problems. However, little is currently know about the capabilities of QAOA, or of the difficulty of the requisite parameters optimization. Here, we study the performance of QAOA on the MaxCut combinatorial optimization problem, optimizing the quantum circuits on a classical computer using automatic differentiation and stochastic gradient descent, using QuantumFlow, a quantum circuit simulator implemented with TensorFlow. We find that we can amortize the training cost by optimizing on batches of problems instances; that QAOA can exceed the performance of the classical polynomial time Goemans-Williamson algorithm with modest circuit depth, and that performance with fixed circuit depth is insensitive to problem size. Moreover, MaxCut QAOA can be efficiently implemented on a gate-based quantum computer with limited qubit connectivity, using a qubit swap network. These observations support the prospects that QAOA will be an effective method for solving interesting problems on near-term quantum computers. The Quantum Approximate Optimization Algorithm (QAOA) is designed to run on a gate model quantum computer and has shallow depth. It takes as input a combinatorial optimization problem and outputs a string that satisfies a high fraction of the maximum number of clauses that can be satisfied. For certain problems the lowest depth version of the QAOA has provable performance guarantees although there exist classical algorithms that have better guarantees. Here we argue that beyond its possible computational value the QAOA can exhibit a form of Quantum Supremacy in that, based on reasonable complexity theoretic assumptions, the output distribution of even the lowest depth version cannot be efficiently simulated on any classical device. We contrast this with the case of sampling from the output of a quantum computer running the Quantum Adiabatic Algorithm (QADI) with the restriction that the Hamiltonian that governs the evolution is gapped and stoquastic. Here we show that there is an oracle that would allow sampling from the QADI but even with this oracle, if one could efficiently classically sample from the output of the QAOA, the Polynomial Hierarchy would collapse. This suggests that the QAOA is an excellent candidate to run on near term quantum computers not only because it may be of use for optimization but also because of its potential as a route to establishing quantum supremacy. In a maximum-r-constraint satisfaction problem with variables x1, x2, ... ,xn , we are given Boolean functions f1, f2, ..., fm each involving r of the n variables and are to find the maximum number of these functions that can be made true by a truth assignment to the variables. We show that for r fixed, there is an integer q ∈ O(log(1 e) e4) such that if we choose q variables (uniformly) at random, the answer to the subproblem induced on the chosen variables is, with high probability, within an additive error of eqr of qr nr times the answer to the original n-variable problem. The previous best result for the case of r = 2 (which includes many graph problems) was that there is an algorithm which given the induced sub-problem on q = O(1 e5) variables, can find an approximation to the answer to the whole problem within additive error en2. For r≥3, the conference version of this paper (in: Proceedings of the 34th ACM STOC, ACM, New York, 2002, pp. 232-239) and independently Andersson and Engebretsen give the first results with sample complexity q dependent only polynomially upon 1 e. Their algorithm has a sample complexity q of O(1 e7). They (as also the earlier papers) however do not directly prove any relation between the answer to the sub-problem and the whole problem as we do here. Our method also differs from other results in that it is linear algebraic, rather than combinatorial in nature.