Hello! My name is Koki Sueda, an internship student.
I have conducted research on the simplification of computation by transforming tensors inspired by quantum computation.
I would like to simplify the 8x8 matrix operation on the CCZ gate of a 3-qubit circuit, which was introduced by Dr. Shinichiro Akiyama at the 3rd Quantum Software Hands-on seminar at the University of Tokyo.
The red-framed part of the images are excerpts from Dr. Akiyama's slide.
Source of red border-boxed excerpts: ：joint-seminar/テンソルネットワークを用いた量子回路シミュレーション.pdf at main · utokyo-qsw/joint-seminar (github.com)
First, for CCZ gates, the following quantum circuit can be represented in an 8 x 8 matrix.
This is decomposed into a product of three different matrices (MPS) of tensors L, C, and R.
A tensor here is a collection of several 2 x 2 matrices according to a certain order.
Generally speaking, there are several ways to contract a tensor, but we will assume that the transformations are performed in the order shown in the following slide.
The tensors L, C, and R can be represented as follows, respectively.
Now, let us define the matrix with the letter on the right shoulder as the transfer matrix, which changes the state of the qubits in the time direction, like the number over the shoulder.
With this transfer matrix in mind, the derivation of the tensor is the key to reducing the number of parameters while retaining all valid components of the parameters.
The quantitative validity can be discussed using entanglement entropy (E.E.), which is inspired by quantum computation.
Let's take a look at the slide in the blue box, an excerpt from the same endowed course at the University of Tokyo, "Application of Tensor Networks in Machine Learning," by Dr. Kenji Harada.
Source of blue boxed excerpts:https://qsw.phys.s.u-tokyo.ac.jp/assets/files/20211207_harada.pdf
Let us estimate the E.E. of the tensor network of CCZ gates according to the definition in the slide below.
For the tensor after MPS, there are at most 4 singular values, since there are at most 2 types of 2 x 2 matrices, combined.
In other words, there are at most ln(4) E.E. ...①.
If we construct a tensor network with one arm, at least 2 degrees of freedom, as in the slide above, the upper limit of E.E is 2ln(2) = ln(4). ... ②.
From ① and ②, we see that the contracted representation of the tensor network of CCZ gates is valid because E.E. does not exceed its upper bound.
The original E.E. was ln(8), but we are able to reduce the parameters while achieving the desired representation by considering entanglement every 2 qubits instead of 3 qubits.
Now, let's look at the tensor-by-tensor calculation.
This time, we consider the inputs α and β for arbitrary states in the range commonly used in Grover's algorithm.
We will perform calculations in two directions: the time direction and the bond dimension direction. For the computation in the time direction, we will do so as defined in the transfer matrix.
For the tensor L,
We can see that the probability of the existence of the first qubit of |0> and |1> one layer below the bond dimension is inherited.
For the tensor C,
Looking at the output in the bond dimension direction, we see that only the term that is the product of PLβ, the |1> component of the first qubit of input, and PCβ, the |1> component of the second qubit of input, is component wise separated from the other possible events.
Furthermore, we can say that tensor C can recursively extract components that are all |1> in its own bond dimension and all inputs in the dimension above it. ... ③.
About Tensor R,
Looking at the output in the Bond dimension direction, we see that the Z-gate operation is performed for the target qubit by the result of any non-zero product of the eight terms, the components of which are all |1> inputs of the qubit in the upper qubit of the tensor R.
This is consistent with the first definition.
As for the output in the time direction, we see that the information αR and βR input to the tensor R is output in the time direction without component inversion for anything other than the Z-gate operation described earlier.
We find that we don't have to compute an 8 × 8 matrix, but can simply use the tensor calculation created by MPS for each one.
The number of parameters is reduced as follows
Before compression: 8 x 8 matrix = 64 parameters
After compression: 32 parameters
The breakdown after compression is as follows.
After compression (tensor L): 4 x 2 parameters
After compression (Tensor R): 4 x 4 parameters
After compression (tensor C): 4 x 2 parameters
This is consistent with the University of Tokyo lecture material showing parameter number reduction in n qubits.
The same operation can be performed from ③ for an n-qubit circuit with multiple tensor C.
Thanks for reading.