Shelby Kimmel

Robust Extraction of Tomographic Information via Randomized Benchmarking

Quantum processing tomography typically reconstructs an unknown quantum dynamical operation by measuring its effects on known states of a quantum device. Taking a different approach of comparing the operation of interest to a set of finite and easily implementable reference operations, a new method can reconstruct any quantum operation reliably.

Super-polynomial quantum speed-ups for boolean evaluation trees with hidden structure

We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth <i>n</i> tree using <i>O</i>(<i>n</i>^2+logω) queries, where ω is independent of <i>n</i> and depends only on the type of subformulas within the tree. We also prove a classical lower bound of <i>n</i>^{Ω(log log <i>n</i>)} queries, thus showing a (small) super-polynomial speed-up.

Entanglement cost of nonlocal measurements

For certain joint measurements on a pair of spatially separated particles, we ask how much entanglement is needed to carry out the measurement exactly. For a class of orthogonal measurements on two qubits with partially entangled eigenstates, we present upper and lower bounds on the entanglement cost. The upper bound is based on a recent result by Berry [Phys. Rev. A 75, 032349 (2007)]. The lower bound, based on the entanglement production capacity of the measurement, implies that for almost all measurements in the class we consider, the entanglement required to perform the measurement is strictly greater than the average entanglement of its eigenstates. On the other hand, we show that for any complete measurement in dxd dimensions that is invariant under all local generalized Pauli operations, the cost of the measurement is exactly equal to the average entanglement of the states associated with the outcomes.

Hamiltonian simulation with optimal sample complexity

We investigate the sample complexity of Hamiltonian simulation: how many copies of an unknown quantum state are required to simulate a Hamiltonian encoded by the density matrix of that state? We show that the procedure proposed by Lloyd, Mohseni, and Rebentrost [Nat. Phys., 10(9):631–633, 2014] is optimal for this task. We further extend their method to the case of multiple input states, showing how to simulate any Hermitian polynomial of the states provided. As applications, we derive optimal algorithms for commutator simulation and orthogonality testing, and we give a protocol for creating a coherent superposition of pure states, when given sample access to those states. We also show that this sample-based Hamiltonian simulation can be used as the basis of a universal model of quantum computation that requires only partial swap operations and simple single-qubit states.Quantum Software from Quantum StatesOne of the hallmarks of quantum computation is the storage and extraction of information within quantum systems. Recently, Lloyd, Mohseni and Rebentrost created a protocol to treat multiple identical copies of a quantum state as “quantum software”, specifying a quantum program to be run on any other state. They use this approach to do principal component analysis of the software state. Here, we expand on their results, providing protocols for running more-complex quantum programs specified by several different states. Our protocols can be used to analyze the relationship between different states (for example, deciding whether states are orthogonal) and to create new states (such as coherent linear combinations of two states). We also outline the optimality of Lloyd et al.’s original protocol, as well as our new protocols.

The quantum query complexity of read-many formulas

The quantum query complexity of evaluating any read-once formula with n black-box input bits is

Quantum adversary (upper) bound

\Theta(\sqrt n)

Robust Calibration of a Universal Single-Qubit Gate-Set via Robust Phase Estimation

. However, the corresponding problem for read-many formulas (i.e., formulas in which the inputs can be repeated) is not well understood. Although the optimal read-once formula evaluation algorithm can be applied to any formula, it can be suboptimal if the inputs can be repeated many times. We give an algorithm for evaluating any formula with n inputs, size S, and G gates using

Experimental Demonstration of a Cheap and Accurate Phase Estimation

O(\min\{n, \sqrt{S}, n^{1/2} G^{1/4}\})

Phase retrieval using unitary 2-designs

quantum queries. Furthermore, we show that this algorithm is optimal, since for any n,S,G there exists a formula with n inputs, size at most S, and at most G gates that requires

Oracles with Costs

\Omega(\min\{n, \sqrt{S}, n^{1/2} G^{1/4}\})