There is a tremendous interest in developing practical applications for noisy intermediate-scale quantum processors without the overhead required by full error correction. Near-term quantum information processing is especially challenging within the standard gate model, as algorithms quickly lose fidelity as the problem size and circuit depth grow. This has lead to a number of non-gate-model approaches such as analog quantum simulation and quantum annealing. These come with specific hardware requirements that are different than that of a universal gate-based quantum computer. We have previously proposed an approach called the single-excitation subspace (SES) method, which uses a complete graph of superconducting qubits with tunable coupling. Without error correction the SES method is not scalable, but it offers several algorithmic components with constant depth, which is highly desirable for near-term use. The challenge of the SES method is that it requires a physical qubit for every basis state in the computer's Hilbert space. This imposes exponentially large resource costs for algorithms using registers of ancillary qubits, as each ancilla would double the required graph size. Here we show how to circumvent this doubling by leaving the SES and fusing it with a multi-ancilla Hilbert space. Specifically, we implement the tensor product of an SES register holding "data" with one or more ancilla qubits, which are able to independently control arbitrary [Formula: see text] unitary operations on the data in a constant number of steps. This enables a hybrid form of quantum computation where fast SES operations are performed on the data, traditional logic gates and measurements are performed on the ancillas, and controlled-unitaries act between. As example applications, we give ancilla-assisted SES implementations of quantum phase estimation and the quantum linear system solver of Harrow, Hassidim, and Lloyd.
There is a tremendous interest in developing practical applications for noisy intermediate-scale quantum processors without the overhead required by full error correction. Near-term quantum information processing is especially challenging within the standard gate model, as algorithms quickly lose fidelity as the problem size and circuit depth grow. This has lead to a number of non-gate-model approaches such as analog quantum simulation and quantum annealing. These come with specific hardware requirements that are different than that of a universal gate-based quantum computer. We have previously proposed an approach called the single-excitation subspace (SES) method, which uses a complete graph of superconducting qubits with tunable coupling. Without error correction the SES method is not scalable, but it offers several algorithmic components with constant depth, which is highly desirable for near-term use. The challenge of the SES method is that it requires a physical qubit for every basis state in the computer's Hilbert space. This imposes exponentially large resource costs for algorithms using registers of ancillary qubits, as each ancilla would double the required graph size. Here we show how to circumvent this doubling by leaving the SES and fusing it with a multi-ancilla Hilbert space. Specifically, we implement the tensor product of an SES register holding "data" with one or more ancilla qubits, which are able to independently control arbitrary [Formula: see text] unitary operations on the data in a constant number of steps. This enables a hybrid form of quantum computation where fast SES operations are performed on the data, traditional logic gates and measurements are performed on the ancillas, and controlled-unitaries act between. As example applications, we give ancilla-assisted SES implementations of quantum phase estimation and the quantum linear system solver of Harrow, Hassidim, and Lloyd.
There is currently a tremendous interest in developing practical applications of NISQ processors without the overhead required by full error correction. Quantum information processing is especially challenging within the gate model, as algorithms quickly lose fidelity as the problem size and circuit depth grow. This has lead to a number of non-gate-model approaches such as analog quantum simulation and quantum annealing. These approaches come with specific hardware requirements that are typically different than that of a universal gate-based quantum computer. We have previously proposed a non-gate-model approach called the single-excitation subspace (SES) method, which requires a complete graph of superconducting qubits. Like any approach lacking error correction, the SES method is not scalable, but it often leads to algorithms with constant depth, allowing it to outperform the gate model in a wide variety of applications. A challenge of the SES method is that it requires a physical qubit for every basis state in the computer's Hilbert space. This imposes large resource costs for algorithms using registers of ancillary qubits, as each ancilla would double the required graph size. Here we show how to circumvent this doubling by leaving the SES and reintroducing a tensor product structure in the computational subspace. Specifically, we implement the tensor product of an SES register holding ``data" with one or more ancilla qubits. This enables a hybrid form of quantum computation where fast SES operations are performed on the data, traditional logic gates and measurements are performed on the ancillas, and controlled-unitaries act between. As an application we give an SES implementation of the quantum linear system solver of Harrow, Hassidim, and Lloyd.
Current quantum computing architectures lack the size and fidelity required for universal fault-tolerant operation, limiting the practical implementation of key quantum algorithms to all but the smallest problem sizes. In this work we propose an alternative method for general-purpose quantum computation that is ideally suited for such ``prethreshold'' superconducting hardware. Computations are performed in the n-dimensional single-excitation subspace (SES) of a system of n tunably coupled superconducting qubits. The approach is not scalable, but allows many operations in the unitary group SU(n) to be implemented by a single application of the Hamiltonian, bypassing the need to decompose a desired unitary into elementary gates. This feature makes large, nontrivial quantum computations possible within the available coherence time. We show how to use a programmable SES chip to perform fast amplitude amplification and phase estimation, two versatile quantum subalgorithms. We also show that an SES processor is well suited for Hamiltonian simulation, specifically simulation of the Schr\"odinger equation with a real but otherwise arbitrary $n\ifmmode\times\else\texttimes\fi{}n$ Hamiltonian matrix. We discuss the utility and practicality of such a universal quantum simulator, and propose its application to the study of realistic atomic and molecular collisions.
We describe an alternative approach to quantum computation that is ideally suited for today's sub-threshold-fidelity qubits, and which can be applied to a family of hardware models that includes superconducting qubits with tunable coupling. In this approach, the computation on an n-qubit processor is carried out in the n-dimensional single-excitation subspace (SES) of the full 2^n-dimensional Hilbert space. Because any real Hamiltonian can be directly generated in the SES [E. J. Pritchett et al., arXiv:1008.0701], high-dimensional unitary operations can be carried out in a single step, bypassing the need to decompose into single- and two-qubit gates. Although technically nonscalable and unsuitable for applications (including Shor's) requiring enormous Hilbert spaces, this approach would make practical a first-generation quantum computer capable of achieving significant quantum speedup.
Quantum computation with a complete graph of superconducting qubits has been recently proposed, and applications to amplitude amplification, phase estimation, and the simulation of realistic atomic collisions given [Phys. Rev. A 91, 062309 (2015)]. This single-excitation-subspace (SES) approach does not require error correction and is practical now. Previously it was shown how to implement symmetric $n\ifmmode\times\else\texttimes\fi{}n$ unitaries in a single step, but not general unitaries. Here we show that any element in the unitary group U(n) can be executed in no more than three steps, for any n. This enables the implementation of highly complex operations in constant time and, in some cases, even allows for the compilation of an entire algorithm down to only three operations. By using this protocol we show how to prepare any pure state of an SES chip in three steps and also how to compute, for a given SES state $\ensuremath{\rho}$, the expectation value $\mathrm{Tr}(\ensuremath{\rho}O)$ of any $n\ifmmode\times\else\texttimes\fi{}n$ Hermitian observable O in a constant number of steps.
We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method ( Single-Ancilla LCU ) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach ( Analog LCU ) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method ( Ancilla-free LCU ) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.
The AKLT state is the ground state of an isotropic quantum Heisenberg spin-1 model. It exhibits an excitation gap and an exponentially decaying correlation function, with fractionalized excitations at its boundaries. So far, the one-dimensional AKLT model has only been experimentally realized with trapped-ions as well as photonic systems. In this work, we successfully prepared the AKLT state on a noisy intermediate-scale quantum (NISQ) era quantum device. In particular, we developed a non-deterministic algorithm on the IBM quantum processor, where the non-unitary operator necessary for the AKLT state preparation is embedded in a unitary operator with an additional ancilla qubit for each pair of auxiliary spin-1/2’s. Such a unitary operator is effectively represented by a parametrized circuit composed of single-qubit and nearest-neighbor CX gates. Compared with the conventional operator decomposition method from Qiskit, our approach results in a much shallower circuit depth with only nearest-neighbor gates, while maintaining a fidelity in excess of 99.99% with the original operator. By simultaneously post-selecting each ancilla qubit such that it belongs to the subspace of spin-up |↑>, an AKLT state can be systematically obtained by evolving from an initial trivial product state of singlets plus ancilla qubits in spin-up on a quantum computer, and it is subsequently recorded by performing measurements on all the other physical qubits. We show how the accuracy of our implementation can be further improved on the IBM quantum processor with readout error mitigation.