It's been silent on my blog for a while now - the past year has been a busy time for me since I was working on my Master's thesis as well as working as a research assistant at Fraunhofer in parallel. Both were great experiences but it became so time-consuming that I totally lost track of posting any content on this blog!

In April 24 I finished my Master's Thesis in Physics in which I was mainly working on tensor networks. We tried to make DMRG (Density Matrix Renormalization Group) for Quantum Chemistry-like setups more efficient by lifting degeneracies in the symmetry sectors of the tensor network structure. Besides from what I learned on the physics-side of this project, it helped me to foster my fascination for method-development.

My work at Fraunhofer was very rewarding as well - I did not only learn a lot, I also had the chance to contribute in two publications in the area of quantum circuit cutting. In case you are interested, I could write a blog entry for one or both of the papers(:

Recently, I attended the MCQST 2023 Conference on which Lídia del Rio presented the research with her collaborators about quantum thought experiments in a quantum computer. They wrote a whole package to do this and describe the ideas in detail in [1]. It's definitely worth checking out the paper and the package - to make you curious let us look at an illustrative example [1, p.4-10].

Example Setting

Let us consider the following setting (as depicted in the image above): Alice has some two-level quantum system R (e.g. a qubit) in the state written in blue. Thus, the probability of obtaining a=0 in a measurement is 1/3 while the result a=1 will be obtained with a probability 2/3. Depending on the outcome, Bob receives the a system in state |0> (if Alice's result was a=0) or in state |+> ~ |0>+|1> (if Alice's result was a=1). In turn, Bob measures his system in the computational basis and can receive the outcome b=0 or b=1. What conclusions can Bob draw about Alice's measurement outcomes based on his? It is assumed that Bob knows the rules upon which Alice sends him the different systems. Thus, if his outcome is b=0 he cannot make any retrodiction since the outcome b=0 could stem from both possible states |0> and |+>. However, if he measures b=1 he knows that his state must have been in |+> and thus he can retrodict that Alice's outcome must have been a=1. Therefore, in one of both cases Bob can draw a deterministic conclusion about Alice's outcome.

So far so good, at this point I'd like to mention that even though this setup seems to be motivated by the Frauchiger-Renner Thought Experiment, we will not talk about apparent paradoxes or fundamental questions in foundations of quantum mechanics themselves. Instead the setting is supposed to be easy to grasp and can therefore neatly serve the purpose to illustrate how a thought experiment can be formalized in terms of quantum circuits. Hence, we will discuss a tool which can be used for quantum thought experiments in general by using a simple example.

Alice's and Bob's Brains in a Quantum Circuit

Next, we will translate this specific setting as a quantum circuit - by going through the above illustration of the resulting circuit. The first qubit is initialized in the state of Alice's system R. Even though this seems to be the only true quantum system at hand, we will act as if there was an external observer who looks at both Alice and Bob and their respective systems. Imagine you are in the position of this external observer and set the Heisenberg cut at this point: You are classical while both Alice and Bob are quantum (as it is done in Neo-Copenhagen interpretations). Then, one also has to model the "brains"/"memory" of both Alice and Bob. We start with Alice first: we assign a wire of the circuit to Alice's reasoning which basically means that somehow the possible measurement results are stored in this respective qubit. The wire representing her memory is initialized in state |0> and is connected to her system R via a CNOT gate. This means that if the system R was in state |0>, the qubit representing Alices would stay in |0>. However, if R was in |1>, Alice's state of memory would be in |1> as well. This way, one can model different measurement outcomes and also Alice's memory in a unitary manner without explicitly including measurements in the circuits yet. This is necessary since from our external perspective everything about Bob and Alice is considered to be quantum, i.e. must be modelled unitarily. Now, we can look at the third wire: It is again initialized in state |0> and remains in this state if Alice's memory is in state |0>. However, the controlled Hadamard will act on the third wire if Alice's memory is in state |1>, hence it would be turned into state |+>. Thus the controlled Hadamard models the system S which Bob receives - conditioned on Alice's measurement outcome of R. Finally, the last wire is again initialized in |0> and is supposed to model Bob's memory. Exactly as Alice's memory, also the relation between Bob's memory and his system S is modeled via a CNOT gate.

As a result we now have a quantum circuit which represents the setup from above from an external perspective. I think already at this point one can see the beauty of this approach - while one needs quite a lot of sentences to explain the simple setup, it can very easily be grasped by the neat quantum circuit. What is left to do now is to model Bob's reasoning regarding his retrodiction on Alice's outcome. We found that Bob can draw a deterministic conclusion about Alice's measurement outcome if his outcome is b=1, in the other case he cannot draw such a conclusion. How can this be mapped into a quantum circuit?

Modelling Bob's reasoning

In the above circuit we added a couple of additional wires. One set represents the four possible logical inferences in this case, and the last two wires will show what Bob's prediction will be based on the initialized inferences. Let's go through this step by step: There are four possible inferences on the measurement outcomes a and b, but only one of them is assumed to hold, namely (b = 1 -> a=1), which is why only the wire corresponding to this inference is initialized in state |1>. The other three inferences, which are assumed not to hold, are initialized in |0> and since there are control nodes from the nonlocal Toffoli-type on those wires, they will not really contribute as long as one does not change the initialization. Those Toffoli-type gates have two control nodes each as well as a NOT at the lower end. One control node is placed on Bob's memory qubit, acting dependently on the outcome b and represents the antecedent of each possible inference. If the inference of a wire assumes b=1 the corresponding Toffoli node on Bob's memory will be black, while it will be white for b=0. The second control node of each Toffoli gate is black in order to be activated according to which inference is initialized with state |1>. The consequent of those inferences is modeled by the lowest two wires. The NOT of each Toffoli is placed on the corresponding wires representing the consequent. Looking at the Toffoli for the inference b=1 -> a=1, one can see that if Bob's memory is in |1> and simultaneously the wire of the corresponding inference is initialized as |1> as well, the NOT on the lowest wire will turn the respective state to |1> (the prediction wires are initialized in |0>). Thus, Bob's prediction can be read off by the states of the prediction wires. Finally, one can also run this circuit and check its consistency - how?

Consistency Checks

In the above image we have put together all we got so far: Alice's actions from before, as well as bob's actions and his reasoning as discussed right above. The consistency of such a model can be checked by measuring the prediction wires as well as Alice's memory qubit. In this case, the only deterministic inference will show itself if Bob's prediction wire for a=1 will be |1> and this will coincide with Alice's memory being in |1>. For the other case, no inference can be done. This way one can check the consistency of the model and if the results show paradoxical outcomes one knows that something went wrong, that something in the logical reasoning / adopted interpretation of quantum theory is getting problematic. Having everything formalized as a quantum circuit will make the analysis of the issues easier.

Final Remarks

It appears to me that Quantum Circuits are not used here because one expects some computational advantage by running them on a Quantum Computer - instead they are used to neatly formalize subsystems and possible inferences of thought experiments. This way, thought experiments can be made more clear and transparent as well as it is easier to see the problem if the outcomes are not consistent. Therefore, their work shows that quantum circuits have a much broader field of application - it is not only about striving for some kind of quantum advantage for specific decision problems, instead they can also be used to formalize concepts in foundations of quantum mechanics; and this is something I have never thought about before, which is why I am so fascinated by the idea.

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Since the machine learning course I did this semester at Tsinghua university was mainly focused on typical data science applications, I was curious in how far those methods can be applied in physics. Of course it is nothing new that neural networks can in principle also be used for physical applications - however, tensor network methods still seem to be dominant in the field of numerical many body physics. Thus, I decided to dive in a little into the literature about the usage of Restrictive Boltzmann Machines (RBM) in many body physics.

What are RBMs?

Usually, RBMs are used for instance for recommendation tasks (e.g. video recommendations on video platforms) and many more. In general, it is a unsupervised learning technique which makes use of minimizing its "energy". Thus, the intuition behind RBMs is, despite their data science applications, already related to physics: We will see that it is no surprise that "Boltzmann" is part of the name of this method. An RBM utilizes input data, tries to extract meaningful features from it and wants to find the probability distribution over the input. This follows the physical intuition as follows: The RBM is a neural network with two layers, a hidden and a visible layer, where each node can adopt binary values. An example network looks like:

where the x denote the visible nodes, the h the hidden nodes and W denotes the weights between both layers. Note that there are no links in between the nodes of a single layer; this is why these networks are called restricted Boltzmann machines.

The network is governed by a corresponding energy function as:

where we also have the offsets of the single nodes (a for the visible nodes and b for the hidden nodes). Given this energy function, one can determine the probability distribution by the Boltzmann distribution where Z is a partition function, as familiar from classical statistical physics. As usual, the energy is supposed to be minimized, what is done by the learning algorithm of the RBM but this should not explained here since it would go beyond the scope of a brief blog entry. At this point I'd only like to mention that there are some difficulties determining e.g. the partition function (which is intractable in general) and that this requires some sophisticated algortihms. If you're interested in this and how the RBMs work exactly, a neat and far more rigorous introduction into RBMs can be found here.

One side note at this point: RBMs were introduced by Geoffrey Hinton after John Hopfield (a physicist) invented the so-called Hopfield networks which are also such an energy based mechanism, based on the physical intuition of Ising models.

Note that so far we only talked about the RBMs as they are used in data science - despite their physical intuition, they had so far nothing to do with neither quantum mechanics nor the many body problem. This is what comes next.

How can this be linked to condensed matter?

As introduced in [1], an RBM that can represent a quantum many body state would look like this:

In comparison to the previous network we changed the labels from x to σ, where the σ's denote e.g. spin 1/2 configurations, bosonic occupation numbers and so on. For them one has to choose a basis, e.g. the σ^z basis. This configuration can be summarized in the set S. Hence, the visible nodes are the N physical nodes of the system. The M hidden nodes h play the role of auxiliary (spin) variables. The authors describe the understanding of such a neural-network quantum state as follows: "The many-body wave function is a mapping of the N−dimensional set S to (exponentially many) complex numbers which fully specify the amplitude and the phase of the quantum state. The point of view we take here is to interpret the wave function as a computational black box which, given an input manybody configuration S, returns a phase and an amplitude according to Ψ(S)" [1, p.2]. Thus, one gives a certain spin configuration as input and the RBM generates the state, in the following form:

Thus, once can recognize that the form of such a neural-network quantum state adopts a similar form as the aforementioned Boltzmann distribution (exponential of energy function). However, there is additionally a sum over all possible hidden configurations which specifies the full state. After setting a state up in this form, one aim could be to find the ground state corresponding to a certain Hamiltonian, and according to the authors of [1], their RBM method gives decent results for this task!

Similarity to tensor networks

Interesting is, that this framework (even though it appears very different) has some analogous quantities as tensor network states. For example, the representational quality of a neural-network quantum state can be increased by increasing the number of hidden states: Thus, the ratio M/N plays a similar role as the bond dimension of a tensor product state! There are many more similarities, which should not be discussed here but can be found in [3].

Nevertheless, I'd like to mention an important distinction, which is also crucial for tensor network states, because some algorithms (DMRG etc.) can only handle area law states properly. While volume law states have an entanglement entropy which scales with the volume of the partitions of a state, an area law state has a scaling only proportionally with the area of the cut. Area law states can be handled better numerically, because the bond dimension of tensor network states explode for volume law states (more on tensor networks and area law can be found in [4]). According to [2] the difference between area law states and volume law states can be captured in a neat way with RBM states: While volume law states must have full connections between the hidden and physical nodes, an area law state has fewer links - this imposes locality in a sense. The RBM states thus give a neat intuition between the differenece of both kinds of states.

All in all, RBMs seem to be an interesting approach to connect both data science methods and many body physics. It may be that they have strengths which the usual tensor networks approaches lack: for instance, the authors of [2, p.888] claim that it might be possible that RBMs might be able to handle volume law states better than usual tensor network approaches do, which would be of course a major benefit. Since I haven't heard of this approach within the condensed matter framework before, I'm very curious how the importance of this method will evolve in future research!

--- References:

[1] Carleo, Troyer, Solving the Quantum Many-Body Problem with Artificial Neural Networks, arXiv:1606.02318

[2] Melco, Carleo, Carrasquilla, Cirac, Restricted Boltzmann machines in quantum physics, https://doi.org/10.1038/s41567-019-0545-1

[3] Chen, Cheng, Xie, Wang, Xiang, Equivalence of restricted Boltzmann machines and tensor network states, arXiv:1701.04831

Recently, I briefly introduced what circuit cutting is, why it is an advisable thing to do with current quantum hardware (NISQ devices) and what additional costs the cutting is causing. However, I did not go into detail how such a circuit cutting method can look like in detail - this is what this entry will be about. In particular, we will have a look on the circuit cutting procedure proposed by Lowe et. al. [1] in which randomly applied channels are able to cut a circuit.

Identity on Cut Circuits

As mentioned in the previous entry, circuit cutting requires to find a proper identity channel on the cut wires which has reasonably low sampling overhead - the definition of the identity is thus the heart of every circuit cutting procedure. In general, such an identity has the form

where Φ_i is some properly chosen quantum channel. The corresponding cost depends on the value κ

which is the L1 norm of the real coefficients of the identity channel above:

Thus, we see that the main possibility to reduce the sampling overhead is to reduce this value. One possibility with small sampling overhead is the following (however, it is not minimal! The method described in [2] has a lower overhead, but we will not go into detail of this).

The identity channel used in [1] looks like

Here, d=2^k is the dimension of the subspace governing the qubits of the cut. The variable z denotes a Bernoulli random value where z=1 appears with probability d/(2d+1). Later, we will derive this form of the identity channel and will see how this probability and also the expectation E_z emerges. Since, there are two values of z, there are also two quantum channels which can be applied. The first one, Ψ_0, is a measure-and-prepare channel:

The unitaries which are applied on the state prior to measurement have to comprise a 2-design (at least) because otherwise the derivation would not work. Such a design is formed by e.g. Clifford gates but there are many possibilities, one could also rely on approximate designs. Since the form of a quantum channel is not so pictorially, you can see in the following how this channel looks like in "circuit-language":

This means, one applies U^\dagger on the k cut wires, measures in z-basis and retrieves a bitstring y. A state in computational basis corresponding to this bitstring gets initialized and then U is applied. All of this is repeated many times. Note that applying such a circuit in the middle of a larger circuit destroys entanglement of the global state and this is also the reason why cutting requires a lot of sampling (quantified by the sampling overhead): The effects of entanglement in the final result must be regained somehow by repeating the procedure numerous times.

The other channel, Ψ_1, is a simpler one. It is the so-called fully depolarizing channel in which all of the information within the cut (within a sample) is lost:

In practice, the action on the cut part of the circuit is as follows: First one measures the k cut wires in the computational basis. Afterwards one takes a uniformly sampled bitstring x and initializes it on the wires - as in the following circuit snippet:

Guiding through the Derivation

Now, as we have settled the definitions, let's go through the derivation of this identity channel! At first, we need an equation which we shall not prove as this would be more involved (based on Haar measure etc.). It is called Werner Twirling Channel:

This equation is particularly nice because the right hand side is much much simpler than the left hand side, which requires all of the unitaries in the design. At the same time, the right hand side could be quickly written out by hand for e.g. d=1. This will come in handy in deriving the identity channel.

The idea of the proof is to start with one channel Ψ_0 and massage it a little to find an expression of Ψ_1 within it and then massaging it a little further and finding an expression for the identity channel.

Hence, start with the channel Ψ_0:

A lot is going on here, thus let's go through the equalities step by step: From the first to the second line the only thing happening is that we insert an identity (using completeness of the computational basis). Going to the next line, the two scalar factors are swapped and the states with index i can be used to rewrite the expression into a Tr. By exploiting the properties of tensor products and the trace, one can draw outside the sum a partial trace expression in the last line. This shape is nice, because we can recognize the left hand side of the above equation and can simplify the underbraced expression:

Since the expression within the sum does not depend on the index j anymore, this sum merely gives a factor d. In the second line we draw the partial trace factor into the brackets and recognize the Ψ_1 channel! Additionally, the part with the SWAP operator can be simplified as well (you can easily prove this by checking it with a 4x4 SWAP matrix and using a general matrix X). All of this helped us immensely in relating both channels to each other. Reshaping this equation a little gives us:

In the second line, we draw a factor outside in order to retrieve the Bernoulli probabilities we have defined previously and then, by respecting the additional sign, this can be easily rewritten as an expectation value in the last line.

What about the cost?

Now as we have both defined and derived the identity circuit expression, let's relate it to the introducing sentences about the sampling overhead. The value of κ can be easily computed:

As we can see, the sampling overhead is exponential in the number of cut qubits - this is a deficit in practice. Even though, there are slightly better circuit cutting procedures, the overhead always scales exponentially in the number of cut qubits. Although unfortunate, this makes perfect sense intuitively: The cutting destroys part of the quantum properties of the system and these must be reproduced classically (by sampling). Mostly everything quantum which is simulated classically scales exponentially (since the Hilbert space dimension grows exponentially with growing number of particles).

Overall, circuit cutting is an interesting new field in quantum computing which might help to go beyond the capabilities of current NISQ devices - nevertheless, there is always a price to pay and it will become evident in future research whether circuit cutting will be a common method or not.

--- References: [1] Angus Lowe, Matija Medvidović, Anthony Hayes, Lee J. O'Riordan, Thomas R. Bromley, Juan Miguel Arrazola, Nathan Killoran. Fast quantum circuit cutting with randomized measurements. 2022. arXiv:2207.14734