Bethe Ansatz solutions for certain Periodic Quantum Circuits
Introduction
One of the promising approaches to study the quantum materials is to simulate them on the engineered quantum platforms currently under development by several groups. In order for these simulations to have predictive powers the platforms have to be properly calibrated and errors to be mitigated. Only reliable way to do it is to compare the results of simulations with available exact analytic results.
The important step in this direction was recently performed [1] where the spectrum (Fourier transform) of the periodic circuits was studied and impressive accuracy of the extracting the parameters of the circuits was achieved (statistical precision below 10−5 rad). However, those results were achieved only for the simulations in the Hilbert subspace of one particle excitations. It is needless to say that not all the parameters of the circuits could be calibrated within study of this Hilbert subspace.
In general, the increase of the number of excitations quickly leads to the results that are untractable on such a high precision level. The only known exceptions are exactly solvable models (such as one dimensional spin chains [2] or Hubbard model [3]) where the number of non-linear equations describing the spectra grows only linearly with system size. However, the direct comparison of those analytic results with the quantum platforms simulation is not straightforward. The difficulty is that the quantum platforms readily operate with the discrete unitary gates rather than with the continuous Hamiltonian dynamics. The reducing the latter to the former requires the Suzuki–Trotter expansion accuracy of which requires smaller steps and deeper circuits. The depth of the circuits is limited due to the noise and it cannot be increased in perpetuity. This motivates the interest to the models where the discrete quantum circuits themselves remain integrable for arbitrary parameters and not only in the limit of small Trotter steps.
This paper is devoted to the analytic solutions of two such models: (i) periodic quantum circuit model; (ii) Chiral Hubbard periodic quantum circuit. Those integrable models in the limit of small Trotter steps match their Hamiltonian counterparts (the integrability of quantum circuits was first pointed out before [4] for and for a non-unitary circuit somewhat similar to our Chiral Hubbard circuit [5]. No expressions for spectra were given in those references). The symmetries of the periodic quantum circuit model are the same for the Hamiltonian version of spin chains. The symmetries of the Chiral Hubbard periodic quantum circuit is lower than that of the Hamiltonian of the fermionic Hubbard model and they are restored only in the limit of small Trotter steps.
We will obtain the complete Bethe ansatz equation and find the spectrum of the string solutions. We will argue that the direct excitations of the strings from the vacuum states is the most effective way to study the integrable many-body physics on the developing quantum platforms. We will use the coordinate version of Bethe ansatz rather than algebraic Bethe ansatz [6], [7] to keep the derivation a little bit more transparent.
Section snippets
General properties
The content of this section is related to the both models considered in this paper.
A quantum circuit operating on sites is shown schematically on Fig. 1. Each site corresponds to either one qubit () or two qubits (Chiral Hubbard).
The total Hilbert space of the system is the direct product of the Hilbert spaces for different sites: The Hilbert space of one site, , is either two-dimensional with the basis for the Circuit I or the direct product of the two
XXZ circuit
Two dimensional Hilbert space of a single qubit is spanned by the basis . The relevant single qubit raising/lowering operators are We will also use the standard Pauli operators acting in the Hilbert space of single qubit, . To avoid confusion, we will always use subscript to label the coordinate and the superscript to label the corresponding Pauli matrix.
Unitary operator, , acts in the Hilbert space of two neighboring sites
Chiral Hubbard model
To obtain the Hubbard model [11], we need to simulate the additional (“spin”) degree of freedom. To achieve it, we replicate the qubit on each site . Replica labels are analogous (with some subtlety to be displayed later) to the spin degrees of freedom so we will introduce Pauli matrices acting in the replica space of one site. The Hilbert space of site is now four-dimensional , so that inter-site gate analogous to (21), acts in -dimensional Hilbert
Summary
In conclusion, I presented complete analytic theory of certain non-trivial quantum circuits. It is important to emphasize that the spectra of all bound states are expressible in terms of the same limited number of parameters (i.e. the number of possible experiments significantly exceeds the number of control knobs). Therefore, analytic results presented here may help not only calibrate the interaction phase in the quantum gates but also put experimental bounds on the possible effects of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
I am thankful to V.Cheianov, L. Glazman, L. Ioffe, K. Kechedzhi, T. Ren, V. Smelyanskiy, and A. Tsvelik for helpful conversations. I also acknowledge discussions with C. Neill, P. Roushan, and Z. Jiang about the possible experiments with the bound states performed on the modern Google platform.
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