Austin Gilliam’s research while affiliated with Jpmorgan Chase & Co. and other places

In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation and experimental results obtained on real quantum hardware. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems.

Amplitude Amplification -- a key component of Grover's Search algorithm -- uses an iterative approach to systematically increase the probability of one or multiple target states. We present novel strategies to enhance the amplification procedure by partitioning the states into classes, whose probabilities are increased at different levels before or during amplification. The partitioning process is based on the binomial distribution. If the classes to which the search target states belong are known in advance, the number of iterations in the Amplitude Amplification algorithm can be drastically reduced compared to the standard version. In the more likely case in which the relevant classes are not known in advance, their selection can be configured at run time, or a random approach can be employed, similar to classical algorithms such as binary search. In particular, we apply this method in the context of our previously introduced Quantum Dictionary pattern, where keys and values are encoded in two separate registers, and the value-encoding method is independent of the type of superposition used in the key register. We consider this type of structure to be the natural setup for search. We confirm the validity of our new approach through experimental results obtained on real quantum hardware, the Honeywell System Model H0 trapped-ion quantum computer.

Selecting a set of basis states is a common task in quantum computing, in order to increase and/or evaluate their probabilities. This is similar to designing WHERE clauses in classical database queries. Even though one can find heuristic methods to achieve this, it is desirable to automate the process. A common, but inefficient automation approach is to use oracles with classical evaluation of all the states at circuit design time. In this paper, we present a novel, canonical way to produce a quantum oracle from an algebraic expression (in particular, an Ising model), that maps a set of selected states to the same value, coupled with a simple oracle that matches that particular value. We also introduce a general form of the Grover iterate that standardizes this type of oracle. We then apply this new methodology to particular cases of Ising Hamiltonians that model the zero-sum subset problem and the computation of Fibonacci numbers. In addition, this paper presents experimental results obtained on real quantum hardware, the new Honeywell computer based on trapped-ion technology with quantum volume 64.

Grover's Search algorithm was a breakthrough at the time it was introduced, and its underlying procedure of amplitude amplification has been a building block of many other algorithms and patterns for extracting information encoded in quantum states. In this paper, we introduce an optimization of the inversion-by-the-mean step of the algorithm. This optimization serves two purposes: from a practical perspective, it can lead to a performance improvement; from a theoretical one, it leads to a novel interpretation of the actual nature of this step. Specifically, we illustrate how this step is a reflection, which is realized by (a) cancelling the superposition of a general state to revert to the original all-zeros state, (b) flipping the sign of the amplitude of the all-zeros state, and finally (c) reverting back to the superposition state. Rather than canceling the superposition, our approach allows for going forward to another state that makes the reflection easier. We validate our approach on set and array search, and confirm our results experimentally on real quantum hardware.

In this paper we discuss Grover Adaptive Search (GAS) for Constrained Polynomial Binary Optimization (CPBO) problems, and in particular, Quadratic Unconstrained Binary Optimization (QUBO) problems, as a special case. GAS can provide a quadratic speed-up for combinatorial optimization problems compared to brute force search. However, this requires the development of efficient oracles to represent problems and flag states that satisfy certain search criteria. In general, this can be achieved using quantum arithmetic, however, this is expensive in terms of Toffoli gates as well as required ancilla qubits, which can be prohibitive in the near-term. Within this work, we develop a way to construct efficient oracles to solve CPBO problems using GAS algorithms. We demonstrate this approach and the potential speed-up for the portfolio optimization problem, i.e. a QUBO, using simulation. However, our approach applies to higher-degree polynomial objective functions as well as constrained optimization problems.

We present a number of quantum computing patterns that build on top of fundamental algorithms, that can be applied to solving concrete, NP-hard problems. In particular, we introduce the concept of a quantum dictionary as a summation of multiple patterns and algorithms, and show how it can be applied in the context of Quadratic Unconstrained Binary Optimization (QUBO) problems. We start by presenting a visual approach to quantum computing, which avoids a heavy-reliance on quantum mechanics, linear algebra, or complex mathematical notation, and favors geometrical intuition and computing paradigms. We also provide insights on the fundamental quantum computing algorithms (Fourier Transforms, Phase Estimation, Grover, Quantum Counting, and Amplitude Estimation) with complete implementations in code.