Implementation and application of efficient state tomography using matrix product states

[g-white]

Abstract

In order for quantum device hardware to continue to improve, characterisation and benchmarking techniques need to stay ahead of the curve. State tomography is an example of device characterisation, checking that the intended final state at the end of a circuit is the one actually produced. A major limitation, however, is that the number of experiments required to produce the density matrix estimate scales like 3^n.
An alternative approach is to model the final state as a matrix product state (MPS). The MPS formalism offers a different picture of quantum states, scaling with the entanglement (maximum Schmidt rank) of the system instead of the number of qubits. While state vector sizes grow like 2^n, most MPSs are poly(n). Applying techniques to find the MPS, tomography can be performed in a significantly reduced number of steps, and verified a-posteriori. This technique would be an ideal addition to Qiskit Ignis, and could be rigorously benchmarked. Further, because MPSs naturally describe the correlation present in the system, this technique could be used for investigation into relevant physical questions, such as the dynamical spread of quantum information.

Description

Background theory of the technique:
https://www.nature.com/articles/ncomms1147.pdf

Experimental verification (2017) on trapped ion qubits:
https://www.nature.com/articles/nphys4244

Would love to be able to get this up and running on Qiskit. There are two alternative methods to follow; the first designates a set of multi-qubit unitaries to disentangle the MPS and characterise the state. The second involves local measurements and more involved post-processing. Ideally we could get both up and running, but the latter is probably more ideal to reduce the circuit gate-count.

If this were completed early in the hackathon, we could put the tools to use to investigate some physical processes. In particular, since the MPS paradigm naturally separates the qubits into groups of correlated qubits -- which is then verified after the fact -- different MPS models could be explored for different dynamical processes, inferring the speed of quantum information.

Finally, there does not currently exist a validation technique for the extension of this method to matrix product operators (MPOs). This would be an exceedingly useful development, allowing the tomography to be comprehensively extended to mixed states (at this stage the estimate can be made, but not validated).

Members

Ideally people familiar with characterisation techniques and matrix product states. A healthy mixture of people with physics background and computer science would be great.

• @githubhandle
• @githubhandle - Slack: @slackhandle email: example@example.com
• Qiskit Coach: @githubhandle

Deliverable

• Qiskit module of the tomography method (necessary)
• Benchmarking of the technique for circuits larger than the current paradigm (ideally 18 qubits on Johannesburg) (necessary)
• Explore physics problems, such as the speed of correlation (maybe)
• Develop a validation technique for MPO tomography (pipe dream)
• Paper?

GitHub repo

[rraymondhp]

Wow... I also posted a similar topic ... #36

[g-white]

Wow... I also posted a similar topic ...

Less than a minute apart as well haha. They would complement each other well though, because MPSs by nature don't allow for verification of highly entangled systems.

[rraymondhp]

I am not familiar with MPS but would love to learn about it from you tomorrow!

[xkwei1119]

this is an interesting project, if you apply MPS tomography on mixed states, do you expect to over- or under- estimate the state fidelity? my understanding is that MPS is an efficient representation for area-law entangled states, such GHZ states, ground state of 1d gapped systems, etc. it will be interesting to apply MPS tomography on GHZ states and compare with known metrics. I agree extension to MPO is very important and will be very cool.

[g-white]

@xkwei1119 that's an interesting point, I would guess that it over-estimates but the verification procedure helps mitigate any bias like that; I would think that the state tomography on the reductions help but I'm not 100% sure. There should be a lot to explore!

[tjkea5]

This sounds super interesting, I did some work on characterisation with MPSs during my honours research. Worked with the quantum information group at Monash uni in Melbourne so also have some experience with open systems formalisms. I'd certainly be interested in working on this project or even just having a chat!
Taylor Kearney

[AidanGG]

I'll join this project: Aidan Dang, Physics, CS and UX/Web.

[MagdaliniZonnios]

I'm interested in joining this group - Magdalini Zonnios, physicist.

[starktech23]

I'm interested. Samanvay Sharma, Computer Science.

[mf-22]

I'm in. Masahiro Fujii, Computer Science.

[BryceFuller]

I’ll mentor - Bryce Fuller

[AidanGG]

Our repo is found at https://github.com/AidanGG/mps_tomo