# Efficient variational simulation of non-trivial quantum states

### Submission summary

 As Contributors: Wen Wei Ho Arxiv Link: https://arxiv.org/abs/1803.00026v4 Date accepted: 2019-02-25 Date submitted: 2019-02-12 Submitted by: Ho, Wen Wei Submitted to: SciPost Physics Domain(s): Theor. & Comp. Subject area: Quantum Physics

### Abstract

We provide an efficient and general route for preparing non-trivial quantum states that are not adiabatically connected to unentangled product states. Our approach is a hybrid quantum-classical variational protocol that incorporates a feedback loop between a quantum simulator and a classical computer, and is experimentally realizable on near-term quantum devices of synthetic quantum systems. We find explicit protocols which prepare with perfect fidelities (i) the Greenberger-Horne-Zeilinger (GHZ) state, (ii) a quantum critical state, and (iii) a topologically ordered state, with $L$ variational parameters and physical runtimes $T$ that scale linearly with the system size $L$. We furthermore conjecture and support numerically that our protocol can prepare, with perfect fidelity and similar operational costs, the ground state of every point in the one dimensional transverse field Ising model phase diagram. Besides being practically useful, our results also illustrate the utility of such variational ans\"atze as good descriptions of non-trivial states of matter.

### Ontology / Topics

See full Ontology or Topics database.

We deeply apologize for the confusion regarding the relation of our considered state preparation protocol with 'circuit depth', its digital/analog nature etc., and believe it is a misunderstanding caused on our part by certain wording and phrasing.

The state preparation protocol we consider is a hybrid variational quantum-classical approach that utilizes the resources of a quantum simulator and a classical computer in an iterative fashion, in order to produce a nontrivial quantum state. It belongs to a large class of protocols which have at its core a variational working principle, see Sec D of https://arxiv.org/abs/1509.04279 for example for a very general description. Naturally, it bears large similarity to the QAOA used to approximately solve classical optimization problems, which is indeed introduced as a gate-model algorithm as the Referee emphasized. However in our case we want to view the protocol we studied more in the context of a variational approach that can be implemented in either a gate-model setting or an analog simulator setting. In the latter case of analog quantum simulation (for example in a present day non-universal trapped ion simulator or neutral Rydberg atom simulator) for which this variational quantum-classical protocol can be run, the notion of gates is less explicit, which was why we had focused less on the 'circuit-depth' and more on the run-time in the previous versions of our paper.

Needless to say, we were inspired much by the works regarding QAOA, and had thus labeled the protocol we considered 'QAOA_p'. However we believe this naming unfortunately amounted in a large part to the confusion that arose, because as mentioned above, our aim was not to study a gate-model algorithm but rather the state preparation protocol as a general, variational approach. This led us to make the erroneous statement that the 'QAOA' is not a digital quantum algorithm (the QAOA as introduced by Farhi is, which the Referee rightly points out it is; we apologize for this). Furthermore, in our replies, we also mistakenly took the parameter p as the 'circuit-depth' of a QAOA circuit, which might have led to even more confusion when the Referee said we made contradictory statements with regards to the total time and the QAOA circuit depth (the objects we are referring to are presumably different). Lastly, another aspect which might have led to misunderstanding could have been in the specific examples considered of the GHZ state and the critical state of the TFIM. In these cases, the Ising interaction used in the protocol can be decomposed into elementary 2-site gates Z_i Z_{i+1}, in exact fashion as the QAOA as analyzed by [Farhi14, arXiv:1411.4028], which can be implemented in a digital quantum simulator. Here an immediate quantum circuit interpretation is indeed possible. However, more generally, for generic target Hamiltonians whose local terms are not mutually commuting, decomposition of our protocol into local gates is much more complicated, and it is not the aim of the paper to study the cost of this.

In response to these issues, we have accordingly modified our paper. To eliminate the confusion that arose, two of our most notable changes include,
1) not referring to the protocol as 'QAOA' but rather a hybrid variational quantum-classical simulation which we call 'VQCS'. This change is reflected in the title as well as the main text.
2) clarifying the setting in which this algorithm is envisioned to be run on -- either a digital quantum simulator or an analog quantum simulator, and made explicit the quantum resources available that can be implemented.
In the case of a digital quantum simulator where evolution is performed by gates resulting in a quantum circuit for the GHZ and critical Ising cases, we have accordingly quoted the circuit depth in a manner similar to the analysis of [Farhi14]. We have also removed the discussion about the QAOA not being a digital quantum algorithm. In making these minor but important structural changes (the concrete analytical and numerical results of our paper remained unchanged), we believe that our main message is now clearer and we hope that the confusion with regards to the relation to QAOA or circuit depth is now resolved.

We emphasize that we agree with the Referee on the points that
i) the QAOA is introduced in the literature as a digital quantum algorithm;
ii) circuit depths as cost invariants are standard in computer science, and are important in the evaluation of an algorithm cost.
It is just that the main point of our reply and our revision of the manuscript is that we would like to view the approach we consider as a general variational approach and not strictly a gate-model algorithm, and which is implementable in either an analog or digital setting. We thank the Referee very much for bringing this up; we hope we have clarified the matter and hope the paper is now suitable for publication.

### List of changes

1. As mentioned in the reply, to reflect our study of the state preparation protocol as a variational approach that can be implemented in either analog or digital quantum simulator settings (and which is not necessarily a gate-model algorithm) we have called it 'Variational Quantum-Classical Simulation' (VQCS) instead of 'QAOA'. We hope this will reduce the confusion that arose with regard to our previous revisions and reply on the properties of QAOA as introduced by Farhi et al (which is indeed a gate-model algorithm). This change is reflected in the title as well as the main text.

2. To clarify and make explicit the context in which the protocol is envisioned for, we have added a discussion on the platforms that it can be run in (both digital and analog simulator settings), as well as the quantum resources available in both settings, in Sec 3.

3. In the case of the transverse field Ising model when the interactions can be implemented easily in a digital quantum simulator in terms of elementary 2-qubit gates Z_i Z_{i+1} and 1-qubit rotations X_i, and where a quantum circuit interpretation is indeed possible, we have quoted the circuit depth accordingly in a manner similar to that of [Farhi14, arXiv:1411.4028].

4. We have completely the discussion on QAOA being an analog algorithm and on the relations between its circuit depth and complexity; as mentioned above, we view our VQCS approach as a general variational approach for which a gate-model interpretation might not always be possible.

### Submission & Refereeing History

Resubmission 1803.00026v4 on 12 February 2019
Resubmission 1803.00026v3 on 18 January 2019
Submission 1803.00026v2 on 6 November 2018