Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. We present a fermion to qubit mapping that outperforms all previous local mappings in both the qubit to mode ratio and the locality of mapped operators. In addition to these practically useful features, the mapping bears an elegant relationship to the toric code, which we discuss. Finally, we consider the error mitigating properties of the mapping-which encodes fermionic states into the code space of a stabilizer code. Although there is an implicit tradeoff between low weight representations of local fermionic operators, and high distance code spaces, we argue that fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular, when undetectable errors correspond to "natural" fermionic noise. We illustrate this point explicitly both for this encoding and the Verstraete-Cirac encoding.

We examine the problem of determining if a 2-local Hamiltonian is stoquastic by local basis changes. We analyze this problem for two-qubit Hamiltonians, presenting some basic tools and giving a concrete example where using unitaries beyond Clifford rotations is required in order to decide stoquasticity. We report on simple results for n-qubit Hamiltonians with identical 2local terms on bipartite graphs. Our most significant result is that we give an efficient algorithm to determine whether an arbitrary n-qubit XYZ Heisenberg Hamiltonian is stoquastic by local basis changes.Accepted in Quantum 2019-03-29, click title to verify arXiv:1806.05405v3 [quant-ph] 29 Apr 2019 1 It is worthwhile noting that the entrywise non-negativity of the matrix exp(−H/kT ) for a given temperature T does not imply that H is a symmetric Z-matrix. It is simple to generate numerical 4 × 4 cases where G = A T A is a non-negative matrix by taking A as a non-negative matrix, but H = − log G is not a symmetric Z-matrix. Thus when H is a symmetric Z-matrix, the non-negativity of exp(−H/kT ) is guaranteed for all values of kT , while in other cases exp(−H/kT ) might be non-negative only for sufficiently small values of kT .2 [1] points out that for 2-local Hamiltonians one can prove that termwise stoquastic is the same as stoquastic. This is however not true for k > 2-local Hamiltonians generally.

Quantum state tomography via local measurements is an efficient tool for characterizing quantum states. However it requires that the original global state be uniquely determined (UD) by its local reduced density matrices (RDMs). In this work we demonstrate for the first time a class of states that are UD by their RDMs under the assumption that the global state is pure, but fail to be UD in the absence of that assumption. This discovery allows us to classify quantum states according to their UD properties, with the requirement that each class be treated distinctly in the practice of simplifying quantum state tomography. Additionally we experimentally test the feasibility and stability of performing quantum state tomography via the measurement of local RDMs for each class. These theoretical and experimental results demonstrate the advantages and possible pitfalls of quantum state tomography with local measurements.

We examine the problem of determining whether a multi-qubit two-local Hamiltonian can be made stoquastic by single-qubit unitary transformations. We prove that when such a Hamiltonian contains one-local terms, then this task can be NP-hard. This is shown by constructing a class of Hamiltonians for which performing this task is equivalent to deciding 3-SAT. In contrast, we show that when such a Hamiltonian contains no one-local terms then this task is easy, namely we present an algorithm which performs this task in a number of arithmetic operations over R which is polynomial in the number of qubits.