Oracles for Gauss's law on digital quantum computers

Formulating a lattice gauge theory using only physical degrees of freedom generically leads to nonlocal interactions. A local Hamiltonian is desirable for quantum simulation, and this is possible by treating the Hilbert space as a subspace of a much larger Hilbert space in which Gauss's law is not automatic. Digital quantum simulations of this local formulation will wander into unphysical sectors due to errors from Trotterization or from quantum noise. In this work, oracles are constructed that use local Gauss law constraints to projectively distinguish physical and unphysical wave functions in Abelian lattice gauge theories. Such oracles can be used to detect errors that break Gauss's law.

Figure 1

A multiqubit subroutine $\oplus$ for comparing two $n$-bit integers $x$ and $y$ by adding all their corresponding bits modulo two. A (non)zero result for $y\oplus x$ means $x$ is (not) equal to $y$.

Figure 2

A generic adder $A$ for adding two $n$-bit integers “in place.” $A$ is assumed to take an incoming carry bit $c_0$(=0,1), whose value can be added to $y+x$ at no additional cost. The overflow bit $h=0$ is needed to express the $(n+1)$-bit sum.

Figure 3

A query to the oracle for a U(1) or ${\mathbb{Z}}_{2^n}$ gauge theory in 1D. (a) The circuit for checking Gauss's law, which accommodates one Dirac fermion. (b) A subtractor routine $S$, which may be thought of as a modified version of $A$.

Figure 4

A schematic summary of how the 1D Gauss law circuit (Fig. 3) acts on a basis state.

Figure 5

A query to the oracle a for U(1) gauge theory in 2D. This example accommodates one flavor of Dirac fermion via the occupation numbers $\nu$ and $p$. To modify this for $g={\mathbb{Z}}_{2^n}$, the overflow bits $h^{\text{OUT,IN}}$ would be removed.

Figure 6

A query to the oracle for a U(1) gauge theory in 3D. This example accommodates one flavor of Dirac fermion via the occupation numbers ${\nu }^{1,2}$ and $p^{1,2}$. To modify this for $g={\mathbb{Z}}_{2^n}$, the four overflow bits $h^{\left\{\text{OUT,IN}\right\},\{1,2\}}$ would be removed.