 # Cutoff Dimension

I see that the concept of cutoff dimension, which is absent in superconducting quantum computing model, adds much more flexibility for interesting computation.

Is there a documentation I can read to understand the concept?

Thanks.

Hey @sophchoe!

A great resource to check out is the theory pages of the Strawberry Fields documentation. The introduction to quantum photonics section should be useful to get an understanding of the cutoff dimension, which relates to the maximum number of photons being simulated. This review article from a few years back may also help introduce some of the language we use in photonic quantum computing.

Thanks,
Tom

Thank you, Tom!

My understanding from “Gaussian Quantum Information” is that any thermal state can be represented by an infinite linear combination of the number base density matrices in an infinite-dimensional Hilbert space. Am I correct in assuming the cutoff dimension would be the number of number bases that we select of that infinite representation?

In physical implementation, the difference between the qubit system and the photonic system, it seems to me, is

• Each qubit is realized with one electron
• Each q-mode is realized with multiple photons.

Is this a correct understanding?

Thank you.

Hi @sophchoe.

Yes, as well as the thermal state in Eq. (27) of Gaussian Quantum Information, you can see the expansion of other common photonic states into the number/Fock basis. For example, coherent states in Eq. (29) and squeezed states in Eq. (33). Although these states are expanded in an infinite-dimensional basis, you can see the coefficients for each Fock state |n\rangle decrease exponentially with n, so that we can introduce an effective cutoff n_{\rm cutoff} and only consider expanding up to n \leq n_{\rm cutoff}.

In physical implementation, the difference between the qubit system and the photonic system, it seems to me, is

• Each qubit is realized with one electron
• Each q-mode is realized with multiple photons.

Right, one nice way to imagine a qumode is simply as a waveguide along a chip (and is indeed how we implement them on our hardware). Waveguides can be interacted individually or together, and can contain multiple photons in each laser pulse. We then measure how many photons come out from the chip.

Haha, good enough approximation. Get it! it’s like

• Approximating a function with a Taylor expansion (if we can tolerate the error, we don’t need an infinite sum) or

• Principal component analysis

• SVD.

As for the photonic model, I say “Neat! Pretty neat!” Can I view a qumode as a communication channel? This to me is a better implementation of the statistical nature of quantum mechanics than the one-to-one qubit model.

I just read your hardware page. Intriguing! Truly intriguing! From the paper, they talk about manipulating electromagnetic field for computation. Is there a snippet page on how you do that?

Thank you so much, Tom!

Hi @sophchoe,

Can I view a qumode as a communication channel?

I’d interpret light running along the waveguide, and potentially interacting with the environment (e.g., during bends in the waveguide), as a channel.

From the paper, they talk about manipulating electromagnetic field for computation. Is there a snippet page on how you do that?

I’m not sure if you’re referring to this paper, but I’d recommend checking it out if you haven’t seen it. Our hardware papers are another good resource.

Hi Tom,

Currently, how many photons do you use per qumode?

In the qubit model, they have to run multiple shots to get the probability of their measurement result.

In your model, theoretically one shot suffices since that statistical process is built in by having multiple photons in a qumode?

Hi @sophchoe!

Currently, how many photons do you use per qumode?

On simulators, the cutoff dimension is a free parameter. It should be large enough so that we are still keeping track of a significant proportion of amplitudes in the Fock basis. This choice depends on the circuit: the presence of squeezers, displacers and other particle generating gates can alter the mean photon number in the circuit. We can hence aim to find the mean photon number at the output of the circuit and adjust the cutoff accordingly.

On hardware, we don’t have a cutoff dimension and there is more freedom on the number of photons. However, there may be some limitation on the detector side: if a single detector is hit by 10+ photons then resolution becomes harder.

In the qubit model, they have to run multiple shots to get the probability of their measurement result.

In your model, theoretically one shot suffices since that statistical process is built in by having multiple photons in a qumode?

In the photonic model, one also has to perform multiple shots to recover the probability of measuring a given pattern of photons. You can see what the output of sampling from a photonic device looks like here. The main difference between qubit and photonic models in this regard is simply that the qubit approach samples from a discrete binary distribution and the photonic approach samples from a discrete distribution over the set of natural numbers.

Thank you, Tom!

“Quantum circuits with many photons on a programmable nanophotonic chip” is a huge help in understanding. https://arxiv.org/pdf/2103.02109.pdf

In the following diagram, could you explain the fixed U(2) transformation part? The rest is very clear from the paper.

Thank you.

Hi @sophchoe,

The U(2) transformation denotes a fixed 50:50 beamsplitter. When equal squeezing on two modes is combined with such a beamsplitter, the result is known as two-mode squeezing.

The reason we have two-mode squeezing is due to hardware considerations - we have four physical waveguides that each contain two-mode squeezed states (e.g., mode 0 and 4 in the diagram above are one physical mode).

I thought beam splitters and phase shifters were included in the U4 transformation. Then would U4 include phase shifters, displacement gates, and Kerr gates instead?

The U4 in the diagram above is physically composed of controllable Mach-Zehnder interferometers and rotation gates, as you can see in the compilation discussion here.

Being programmable, the array of MZ-interferometers and rotations can realize an arbitrary 4-mode interferometer. Such a transformation does not include displacement and Kerr gates.

We separate out the 50:50 beamsplitters of the two-mode squeezing from the rest of the interferometer section out of simplicity, since those beamsplitters are fixed and would also couple the two U4 interferometers.

Hi Tom,

Is your standard mode of measurement Homodyne detection? Would it be safe to think that’s the way of counting the number of photons per qumode?

Hi @sophchoe,

In general some standard measurement types are:

1. Homodyne (measuring in position or momentum basis)
2. Photon number counting (project into Fock basis)
3. Heterodyne (project onto a coherent state)

For our X8 chip, we use PNR (photon number resolving) detectors to count the number of photons coming out of each mode.

Thank you for the clarification! Greatly helpful!

Currently, is X8 for up to 8 qumodes?

No problem!

Yes, X8 an 8 mode device with a fixed circuit structure (see here). The 8 modes are generated through 4 physical waveguides. Each waveguide contains a pair of modes that can be thought of as being squeezed by the same amount and then passed through a 50:50 beamsplitter.

In practice, you don’t have to use all of the 8 modes.

Yes, I see that. Just 2-mode computations are creating excellent results.

Thank you for all your feedback!

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Is it a correct understanding that
“Measurement (projection of each qumode result to Fock basis) is being performed per qumode. That means each qumode is represented in its own Fock space.”?

Yes, we do photon number counting with the X8 chip, which projects each qumode onto a Fock state (corresponding to the number of photons counted in the detector).

Before a qumode gets measured and projected into the Fock basis, one can in principle describe the qumode in any basis. For example, a squeezed state is a superposition over position eigenstates (though it could also be expanded in another basis). When the squeezed state hits the photon number detectors, we project into the Fock basis.

would it be

• before the measurement, the state of each qumode is represented by an infinite sum of different number basis states (approximated by the defined cutoff dimension)
• after the measurement, you observe the superposition of each qumode collapsing into one of the number basis state?