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arXiv:1904.10440 · Physical Review A

Minimally entangled state preparation of localized wave functions on quantum computers

Natalie Klco, Martin J. Savage A

Published Jul 17, 2020• 67 citations
arXivHTMLPDF

Abstract

Initializing a single site of a lattice scalar field theory into an arbitrary state with support throughout the quantum register requires $O({2}^{{n}_{Q}})$ entangling gates on a quantum computer with ${n}_{Q}$ qubits per site. It is conceivable that instead initializing to functions that are good approximations to states may have utility in reducing the number of required entangling gates. In the case of a single site of a noninteracting scalar field theory, initializing to a symmetric exponential wave function requires ${n}_{Q}\ensuremath{-}1$ entangling gates, the minimal number necessary to create an entangled state of all ${n}_{Q}$ qubits. This is compared with the ${2}^{{n}_{Q}\ensuremath{-}1}+{n}_{Q}\ensuremath{-}3+{\ensuremath{\delta}}_{{n}_{Q},1}$ required for a symmetric Gaussian wave function. In this work, we explore the initialization of one-site (${n}_{Q}=4$), two-site (${n}_{Q}=3$), and three-site (${n}_{Q}=3$) noninteracting scalar field theories with symmetric exponential wave functions using IBM's quantum simulators and quantum devices (Poughkeepsie and Tokyo). With the digitizations attainable with ${n}_{Q}=3,4$, these tensor-product wave functions are found to have large overlap with a Gaussian wave function and provide a suitable low-noise initialization for improvement and Somma inflation. In performing these simulations, we have employed a workflow that interleaves calibrations to mitigate systematic errors in production. The calibrations allow tolerance cuts on gate performance including the fidelity of the symmetrizing Hadamard gate, both in vacuum (${|\mathbf{0}\ensuremath{\rangle}}^{\ensuremath{\bigotimes}{n}_{Q}}$) and in medium (${n}_{Q}\ensuremath{-}1$ qubits initialized to an exponential function). The results obtained in this work are relevant to systems beyond scalar field theories, such as the deuteron radial wave function, two- and three-dimensional Cartesian-space wave functions, and nonrelativistic multinucleon systems built on a localized eigenbasis.

DOI
10.1103/physreva.102.012612

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Quantum computing promises to provide exponential speed-ups to certain classes of problems. In many such quantum algorithms, a classical vector b\mathbf{b}b is encoded in the amplitudes of a quantum state ∣b⟩| b\rangle∣b⟩. However, efficient amplitude encoding of an arbitrary vector b\mathbf{b}b in ∣b⟩| b \rangle∣b⟩ is known to be a difficult problem that can reverse the gains of such algorithms. More importantly, an arbitrary quantum state of QQQ qubits generally requires ∼2Q\sim 2^Q∼2Q number of entangling gates, which is problematic for today's Noisy-Intermediate Scale Quantum (NISQ) computers where large numbers of such error-prone gates will result in significant decoherence before initialization is completed. In this work, we demonstrate a deterministic and scalable initialization algorithm that allows for states with low entanglement to be initialized on actual quantum computers with more than an order of magnitude less entangling gates and significantly shallower circuits as compared to isometric decomposition. For states with higher entanglement, optimally approximate states with lower entanglement may be found and initialized with similar performance. We show this to be true for various cases of interest such as the normal and log-normal distributions. Unlike variational approaches, our method requires no classical optimizations in high dimensional spaces and is not plagued with exponentially vanishing gradients or a multitude of local minimas.
Renormalization group ideas and effective operators are introduced to efficiently prepare ground states of massive lattice field theories on digital quantum devices. This is accomplished with a systematic approximation through localized unitaries that removes an exponentially costly barrier in the spatial volume of the quantum simulation. With these methods, classically computed ground states in a spatial volume LLL, containing a few Compton wavelengths, can be used to determine operators for preparing the ground state toward the thermodynamic limit with a precision improving as ${e}^{\ensuremath{-}mL}$ on beyond-classical quantum registers. Due to the exponential spatial decay of correlations in massive theories and the double exponential suppression of digitization artifacts in the number of qubits representing the scalar field, the derived fixed-point quantum circuits are expected to be relevant for simulations of quantum field theories throughout the evolution from small-scale near-term quantum devices to large-scale fault-tolerant quantum computers.
An ultimate goal of quantum computing is to perform calculations beyond the reach of any classical computer.It is therefore imperative that useful quantum computers be very difficult to simulate classically, otherwise classical computers could be used for the applications envisioned for the quantum ones.Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits N or the depth D of the circuit.This difficulty has triggered recent experiments on deep, random circuits that aim to demonstrate that quantum devices may already perform tasks beyond the reach of classical computing.These real quantum computing devices, however, suffer from many sources of decoherence and imprecision which limit the degree of entanglement that can actually be reached to a fraction of its theoretical maximum.They are characterized by an exponentially decaying fidelity F ∼ ð1 -ϵÞ ND with an error rate ϵ per operation as small as ≈1% for current devices with several dozen qubits or even smaller for smaller devices.In this work, we provide new insight on the computing capabilities of real quantum computers by demonstrating that they can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer.Our algorithms compress the representations of quantum wave functions using matrix product states, which are able to capture states with low to moderate entanglement very accurately.This compression introduces a finite error rate ϵ so that the algorithms closely mimic the behavior of real quantum computing devices.The computing time of our algorithm increases only linearly with N and D in sharp contrast with exact simulation algorithms.We illustrate our algorithms with simulations of random circuits for qubits connected in both one-and twodimensional lattices.We find that ϵ can be decreased at a polynomial cost in computing power down to a minimum error ϵ ∞ .Getting below ϵ ∞ requires computing resources that increase exponentially with ϵ ∞ =ϵ.For a two-dimensional array of N ¼ 54 qubits and a circuit with control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours.For more complex gates such as a SWAP gate followed by a controlled rotation, the error rate increases by a factor 3 for similar computing time.Our results suggest that, despite the high fidelity reached by quantum devices, only a tiny fraction ð∼10 -8 Þ of the system Hilbert space is actually being exploited.