Quantum algorithms for weighted constrained sampling and weighted model counting

Abstract We consider the problems of weighted constrained sampling and weighted model counting, where we are given a propositional formula and a weight for each world. The first problem consists of sampling worlds with a probability proportional to their weight given that the formula is satisfied. The latter is the problem of computing the sum of the weights of the models of the formula. Both have applications in many fields such as probabilistic reasoning, graphical models, statistical physics, statistics, and hardware verification. In this article, we propose quantum weighted constrained sampling (QWCS) and quantum weighted model counting (QWMC), two quantum algorithms for performing weighted constrained sampling and weighted model counting, respectively. Both are based on the quantum search/quantum model counting algorithms that are modified to take into account the weights. In the black box model of computation, where we can only query an oracle for evaluating the Boolean function given an assignment, QWCS requires $$O(2^{\frac{n}{2}}+1/\sqrt{\text {WMC}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mtext>WMC</mml:mtext> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> oracle calls, where n is the number of Boolean variables and $$\text {WMC}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>WMC</mml:mtext> </mml:math> is the normalized between 0 and 1 weighted model count of the formula, while a classical algorithm has a complexity of $$\Omega (1/\text {WMC})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mtext>WMC</mml:mtext> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . QWMC takes $$\Theta (2^{\frac{n}{2}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> oracle calss, while classically the best complexity is $$\Theta (2^n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Θ</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , thus achieving a quadratic speedup.