Abstract
The Hidden Weighted Bit function plays an important role in the study of classical models of computation. A common belief is that this function is exponentially hard for the implementation by reversible ancilla-free circuits, even though introducing a small number of ancillae allows a very efficient implementation. In this paper, we refute the exponential hardness conjecture by developing a polynomial-size reversible ancilla-free circuit computing the Hidden Weighted Bit function. Our circuit has size $O(n^{6.42})$, where $n$ is the number of input bits. We also show that the Hidden Weighted Bit function can be computed by a quantum ancilla-free circuit of size $O(n^2)$. The technical tools employed come from a combination of Theoretical Computer Science (Barrington's theorem) and Physics (simulation of fermionic Hamiltonians) techniques.
Highlights
The origins of the Hidden Weighted Bit function go back to the study of models of classical computation. This function, denoted HWB, takes as input an n-bit string x and outputs the k-th bit of x, where k is the Hamming weight of x; if the input weight is 0, the output is 0. It is best known for combining the ease of algorithmic description and implementation by classical Boolean circuits with the hardness of representation by Ordered Binary Decision Diagrams (OBDDs) [1]—a popular tool in VLSI CAD [2]
The difference between complexities of logarithmic-depth implementations of HWB by circuits and an exponential lower bound for the size of the OBDD [3] is a startling two exponents
We highlight that removing the constraint that the order of variables in the OBDD is fixed allows implementing HWB as a polynomial-size BDD [5]
Summary
The origins of the Hidden Weighted Bit function go back to the study of models of classical computation. This allows us to employ Barrington’s theorem [5] to implement the gates C5|Mi (f (x\B); B) in an ancilla-free fashion by expressing them as polynomial-size branching programs with the input x\B and computing into B Each instruction in such program realizes a permutation of 5-bit strings controlled by a single bit and it can be mapped into a reversible circuit over 6 = 5+1 wires. The n-bit hwb function can be implemented by an ancilla-free circuit with log(n) + 1 layers of C-type gates. The n-bit hwb function can be implemented by an ancilla-free reversible circuit of size O(n6.42). We can order the factors in such a way as to allow massive cancellation between consecutive CNOT circuits Cpp and implement the first product with just O(n2) total gates
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