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NISQ Algorithm for Hamiltonian Simulation via Truncated Taylor Series
by Jonathan Wei Zhong Lau, Tobias Haug, Leong Chuan Kwek, Kishor Bharti
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Submission summary
Authors (as registered SciPost users):  Jonathan Wei Zhong Lau · HaiRui Wei 
Submission information  

Preprint Link:  scipost_202105_00027v2 (pdf) 
Date submitted:  20211012 10:24 
Submitted by:  Lau, Jonathan Wei Zhong 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Simulating the dynamics of manybody quantum systems is believed to be one of the first fields that quantum computers can show a quantum advantage over classical computers. Noisy intermediatescale quantum (NISQ) algorithms aim at effectively using the currently available quantum hardware. For quantum simulation, various types of NISQ algorithms have been proposed with individual advantages as well as challenges. In this work, we propose a new algorithm, truncated Taylor quantum simulator (TQS), that shares the advantages of existing algorithms and alleviates some of the shortcomings. Our algorithm does not have any classicalquantum feedback loop and bypasses the barren plateau problem by construction. The classical part in our hybrid quantumclassical algorithm corresponds to a quadratically constrained quadratic program (QCQP) with a single quadratic equality constraint, which admits a semidefinite relaxation. The QCQP based classical optimization was recently introduced as the classical step in quantum assisted eigensolver (QAE), a NISQ algorithm for the Hamiltonian ground state problem. Thus, our work provides a conceptual unification between the NISQ algorithms for the Hamiltonian ground state problem and the Hamiltonian simulation. We recover differential equationbased NISQ algorithms for Hamiltonian simulation such as quantum assisted simulator (QAS) and variational quantum simulator (VQS) as particular cases of our algorithm. We test our algorithm on some toy examples on current cloud quantum computers. We also provide a systematic approach to improve the accuracy of our algorithm.
List of changes
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Reports on this Submission
Report #2 by Anonymous (Referee 4) on 20211125 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202105_00027v2, delivered 20211125, doi: 10.21468/SciPost.Report.3904
Report
Overall, the authors took great efforts to answer my questions and to address my comments. My main criticism was related to the dimension of the variational space (i.e. the number of cumulative $k$moment states) required to faithfully describe the timeevolved state $\psi(t)\rangle$. To some extent, I do agree with the authors that this is related to the fundamental question of expressibility of variational quantum circuits. I would not go so far and say that this is in general a completely open question for all variational quantum algorithms. Most likely it is true when it comes to time evolution of many body systems. On the other hand, if one (for instance) uses a variational quantum eigensolver to prepare ground states of local gapped Hamiltonians, we know that these states should fulfil the area laws of entanglement and are thus described by finite bonddimension tensor networks. According to that, we know that such states are described by a number of parameters that only scale polynomially in the system size, in which case a "relatively" short depth quantum circuits should provide sufficient expressibility. Time evolution as discussed in this manuscript is of course a different story. Nevertheless, I would add the statement that this fundamental problem of expressibility does particularly emerge in variational algorithms for time evolution.
I have the feeling that my question of how quickly the required size of the $k$moment stateset grows as a function of time has not really been answered. The authors say that in the worst case, the parameter $K$ is equal to the rank of the Hamiltonian, but that means that in this worst case the method is impractical. I do not understand the statement of the authors: "... Thus, the number of states that we require in our Ansatz scales linearly with the rank of the Hamiltonian. We believe that this scales favourably compared to other NISQ algorithms such as VQS and VFF." How can this clearly exponential scaling be favourable compared to something else?
I still have the feeling that in the cases where the numerical results match the exact time evolution (for example Figure 3 a)), the number of basis states matches (or exceeds) the Hilbert space dimension. The authors pointed out that in this case there are 137 states in the set while the \textit{full} Hilbert space dimension is $2^8=256$. But of course the Ising model studied here exhibits certain symmetries, like reflection around the center or a global $Z_2$symmetry which is perhaps (?) satisfied by the ansatz. These symmetries might easily reduce the dimension by a factor of 2.
The authors draw several connections to Krylov time evolution. In these algorithms the task is to apply $e^{i \Delta t H}$ at each time step to the current state. To this end one constructs the Krylov subspace at every time step, based on powers of $H$ applied to the state. The number of Krylov vectors is related to the size of the time step one is able to perform. It seems to me that the algorithm proposed by the authors performs a single Krylovtimestep from the initial state. Thus, the maximum time that can be reached might be very limited.
To summarize, I am still concerned that the authors are overselling their approach to a certain extent. In my opinion, it is very important to point out clearly the capabilities of proposed quantum algorithms. In particular to elaborate on perspectives what these algorithms are able to achieve what classical computers cannot. For me, in the present case the answer is: With this algorithm one can time evolve a highly entangled state (a state that cannot be stored efficiently on a classical computer) for a short amount of time. The quantum device has no role here if the initial state is a product state. In this case, everything becomes classical. This point should be communicated very clearly.
Requested changes
I would ask the authors to point out clearly the framework in which their algorithm has meaningful applications. In my opinion this is the following: If a quantum device prepares a highly entangled state, i.e. a state that is difficult to store classically, this algorithm can be used to evolve such a state for a short period of time.
Alternatively, one could provide a detailed analysis on how many basis states are required as a function of time in order to approximate the state to a given fidelity. At the moment the authors say that in the worst case the number of basis states matches the rank of the Hamiltonian. But at this point, the algorithm is impractical.
Report #1 by Anonymous (Referee 3) on 2021119 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202105_00027v2, delivered 20211109, doi: 10.21468/SciPost.Report.3818
Report
The authors have addressed most of my comments and have improved on the previous version of the manuscript by fixing the bigger drawbacks. However, I think there is still room for better clarity and completeness.
1) The response to point 2 in my previous report is not reflected in any changes in the manuscript.
2) The authors have not replied to point 5 in my previous report.
3) A similar comment to the previous point applies to the results in fig 5. K = 3 performs remarkably well, this is a feature that should be discussed.
4) The clarity of fig. 5 could be improved by using different markers for different lines.
5) Some formal inaccuracies that I pointed out in the previous version still persist, please fix them.
6) Typo in the generalised eigenvalue equation.
7) A comment is misplaced above eq. (23).