Introduction
Latent interest in the subject of quantum computing was awoken during an evening lecture by IBM. After reading Quantum computing for the very curious, we decided to take on the subject. The initial objective was to search quantum circuit space to find a circuit that implements the Toffoli gate.
Let's be clear about it: we did not meet that objective. Quantum circuit space is way too vast to be explored using classical computing by brute force alone. Hence the hype. And we were missing the knowledge and intuition to compensate for that fact using heuristics or other shortcuts.
However, we did learn how to create the matrix representation of a quantum circuit. That is the essence of this post, on which the brute force search is based. The section Matrix Representations provides a formal summary.
In addition, the brute force technique developed helped us in gaining insight and making interesting observations in the domain. And we were able to find a quantum circuit for some simpler problems, such as controlled Pauli gates and the simplest quantum Fourier transform.
Information Sources
We will provide references wherever we feel they are relevant. However, introductions that we found useful include:
- Quantum computing for the very curious, as said before, the main trigger for this exercise, and a detailed introduction to the subject. Beware, a first reading will take a couple of hours to complete. Also, the article does not focus too much on the matrix representation that is the main focus of this post.
- A field guide to quantum computing, by IBM, an excellent introduction to the subject. It covers the subject with more speed, and consequently takes you a lot further. The material is accompanied by OpenQasm and Qiskit examples, to try things out. Note however that we will not be using any of these in the context of this post.
- Wikipedia's page on quantum logic gates and many other Wikipedia pages related to quantum computing. The information on the pages is of high quality, and after reading the former sources, can be digested without too many obstacles.
Prerequisites
If you are not familiar with quantum computing and the concepts and notations of the domain, we invite you to at least skim one of the sites mentioned under Information Sources first.
We encourage you not to be scared off by the rather mathematical notation, as the basic concepts remain fairly simple. If you are familiar with complex numbers and have a basic knowledge of matrices, you should be fine.
Matrix Representations
We provide a formal overview of how to create the matrix representation of a quantum circuit and its parts here. We will make a strict distinction between quantum entities and their matrix representations, and use \({\oldstyle M}(...)\) to denote the matrix representation of any quantum entity, including quantum states, gates, circuits and circuit layers.
In practice, some quantum entities and their matrix representations are often denoted using the same symbol. With the exception of this section, we will also take that liberty throughout this post.
Computational basis states of a qubit: The computational basis states of a single qubit are represented by 2x1 matrices, vectors. $${\oldstyle M}(|0\rangle)=\begin{bmatrix}1\\0\end{bmatrix}, {\oldstyle M}(|1\rangle)=\begin{bmatrix}0\\1\end{bmatrix}$$ Here, \(|0\rangle\) and \(|1\rangle\) are the computational basis states of a single qubit.
The quantum state of a qubit: The quantum state of a qubit is expressed as a linear combination of the computational basis vectors with complex coefficients and length 1, represented as a vector with the coefficients of that linear combination. $${\oldstyle M}(|\psi\rangle)={\oldstyle M}(\alpha|0\rangle+\beta|1\rangle)=\begin{bmatrix}\alpha\\\beta\end{bmatrix}$$ Here, \(|\psi\rangle\) is the quantum state of a single qubit.
Computational basis states of a collection of qubits: The computational basis states of a collection of qubits are represented by 2\(^N\)x1 matrices, vectors, defined as the tensor product of the computational basis state representation for each qubit in the collection. $${\oldstyle M}(|q_1q_2...q_N\rangle)={\oldstyle M}(|q_1\rangle)\otimes{\oldstyle M}(|q_2\rangle)...{\oldstyle M}(|q_N\rangle)$$ Here, \(|q_i\rangle, i=1...N\) represent computational basis states of the individual qubits. That is, \(q_i, i=1...N\) equal 0 or 1.
The quantum state of a collection of qubits: The quantum state of a collection of qubits is expressed as a linear combination of the computational basis vectors with complex coefficients and length 1, represented as a vector with the coefficients of that linear combination. It can be written as the tensor product of the vectors representing the quantum state of the individual qubits. $${\oldstyle M}(|\psi_1\psi_2...\psi_N\rangle)={\oldstyle M}(|\psi_1\rangle)\otimes{\oldstyle M}(|\psi_2\rangle)...{\oldstyle M}(|\psi_N\rangle)$$ Here, \(|\psi_1\psi_2...\psi_N\rangle\) represents the quantum state of the collection of qubits and \(|\psi_i\rangle, i=1...N\) represent the quantum states of the individual qubits.
You will not be using this rule very often. You will typically start with one of the computational basis states and then travel through a quantum circuit using the rules below.
Single wire quantum gates: Any single wire quantum gate operating on a single qubit can be represented by a unitary 2x2 matrix with complex elements. All quantum operations must be linear, so it suffices to describe the function on each one of the basis states and let the mixed states be defined by linearity. $${\oldstyle M}(G|0\rangle)=\begin{bmatrix}\alpha\\\beta\end{bmatrix},{\oldstyle M}(G|1\rangle)=\begin{bmatrix}\gamma\\\delta\end{bmatrix} \Rightarrow {\oldstyle M}(G)=\begin{bmatrix}\alpha&\gamma\\\beta&\delta\end{bmatrix}$$ Here, \(G\) represents a quantum gate, and we use juxtaposition to indicate the application of a gate to a qubit's quantum state.
Single wire quantum gate application: The matrix product is used for the application of a single wire gate to a qubit's quantum state. $${\oldstyle M}(G|\psi\rangle)={\oldstyle M}(G){\oldstyle M}(|\psi\rangle)$$ A series of single wire gates: To apply multiple gates to a qubit, multiply the matrix representation of the gates in reverse order. $${\oldstyle M}(G_1 G_2 ... G_N)={\oldstyle M}(G_N)...{\oldstyle M}(G_2){\oldstyle M}(G_1)$$ Here, \(G_i, i=1...N\) represent gates, and we use juxtaposition to indicate the serial application of multiple gates to the quantum state of a single qubit.
Strictly speaking, this rule is redundant, as it is a special case for a quantum circuit with only one wire, and as such it is covered by the more generic matrix representation rules following below. In any case, it should not be used when deriving the matrix representation of a quantum circuit with multiple wires.
Parallel single wire gates, layers: To apply a collection of gates to a collection of qubits in parallel, apply the tensor product to the matrices of the gates, in order. $${\oldstyle M}(G_1\parallel G_2...G_N)={\oldstyle M}(G_1)\otimes{\oldstyle M}(G_2)...{\oldstyle M}(G_N)$$ Here, we use the \(\parallel\) operator to indicate the parallel application of multiple single wire gates to a collection of qubits. More precisely, the expression \(G_1\parallel G_2...G_N\) represents a layer in a quantum circuit consisting of \(N\) wires, each wire with its own gate.
The notation with the \(\parallel\) operator is not part of the general notational conventions in quantum computing. However, we need a notation for a layer in a quantum circuit, because it is an intermediate level to determine the matrix representation of a quantum circuit. Hey, quantum computing is all about entanglement, and that's happening between different wires.
Whenever no gate is present in the layer on a wire \(i\), use the identity matrix \(G_i=I\).
Layer application: The matrix product is used for the application of a layer to the quantum state of a collection of qubits. $${\oldstyle M}(L|\psi_1\psi_2...\psi_N\rangle)={\oldstyle M}(L){\oldstyle M}(|\psi_1\psi_2...\psi_N\rangle)$$ Here, \(|\psi_1\psi_2...\psi_N\rangle\) represents the quantum state of the collection of qubits.
A series of layers, quantum circuits: To apply multiple layers to a collection of qubits, multiply the matrix representation of the layers in reverse order. $${\oldstyle M}(L_1 L_2...L_N)={\oldstyle M}(L_N)...{\oldstyle M}(L_2){\oldstyle M}(L_1)$$ Here, \(L_i, i=1...N\) represent layers, and we use juxtaposition to indicate the serial application of multiple layers in a quantum circuit.
Controlled gate, two wires: The matrix for a 2-wire layer consisting of a control gate and a controlled gate is the sum of two tensor products, one for the case where the control qubit state is \(|0\rangle\), and one for the case where the control qubit state is \(|1\rangle\). $${\oldstyle M}(C\parallel C(G))=P_0\otimes I+P_1\otimes{\oldstyle M}(G)$$ Here, \(C\) refers to the control gate and \(C(G)\) refers to the controlled gate \(G\), which is, for clarity, not restricted to the not gate \(X\). The matrices \(P_0\) and \(P_1\) realise the selection on the state of the control qubit and are defined as $$P_0=|0\rangle\langle 0|=\begin{bmatrix}1&0\\0&0\end{bmatrix}, P_1=|1\rangle\langle 1|=\begin{bmatrix}0&0\\0&1\end{bmatrix}.$$ The term control gate is not part of the general terminology in quantum computing, where the correct terms are a (2-wire) controlled gate with a control wire and a target wire. However, splitting the 2-wire controlled gate into two connected single wire entities greatly simplifies notation and implementation. We will refer to them as pseudo-gates.
Controlled gate, more than two wires: The matrix for a layer with more than two wires, containing a single control gate and a corresponding controlled gate, equally is the sum of two tensor products, one for the case where the control qubit state is \(|0\rangle\), and one for the case where the control qubit state is \(|1\rangle\), but in this case the products involve the matrix representations of the gates on other wires. $$ \begin{align} {\oldstyle M}(G_1&\parallel G_2...C_P...C(G_Q)...G_N)=\\ &{\oldstyle M}(G_1)\otimes{\oldstyle M}(G_2)...P_{0,P}...I_Q...{\oldstyle M}(G_N)+\\ &{\oldstyle M}(G_1)\otimes{\oldstyle M}(G_2)...P_{1,P}...G_Q...{\oldstyle M}(G_N) \end{align} $$ Here, \(P,Q=1...N\) with \(P\ne Q\). Furthermore, \(C_P\) means the control gate is on wire \(P\) and \(C(G_Q)\) means the controlled gate is \(G_Q\) on wire \(Q\). Also, \(P_{D,P},D=0,1\) is \(P_D\) in position \(P\) and \(I_Q\) is just the identity matrix in position \(Q\). And it should be clear that \(P\gt Q\) is allowed as well.
Multiple control wires: We did not investigate layers that contain a controlled gate with multiple control wires or multiple controlled gates working independently, but we expect a term is required in the sum of tensor products for all possible combinations of \(P_0\) and \(P_1\) for the different control gates, each term with the appropriate substitute matrix for the corresponding controlled gates.
Multi-wire gates, other than controlled: Also, the matrix representation rules should be extended to support multi-wire gates that are not control gates, such as the swap gate.
Brute Force Search on Quantum Circuit Space
With the rules for the matrix representation of quantum entities at hand, it is fairly simple to implement an abstract data type for a quantum circuit and simulate the quantum circuit. It comes down to the top-down derivation of the matrix representation of the underlying quantum elements and multiplying the matrix for the quantum circuit with the vector representation of the input for the circuit. Java code snippets are available at the end of this post, under the section Simulating a Quantum Circuit.
Note that the quantum state of a qubit is not directly observable. Suppose the quantum state of a qubit is given by $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle.$$ If that qubit is measured (in the standard basis), the result represented by \(|0\rangle\) is obtained with probability \(|\alpha|^2\) and the result represented by \(|1\rangle\) is obtained with probability \(|\beta|^2\), as prescribed by the Born rule. Therefore, strictly speaking, we do not simulate the quantum circuit. We merely calculate the probability amplitudes of the computational basis for the output of the quantum circuit.
Brute force search on quantum circuit space is then about finding a way to iterate over all possible quantum circuits, preferably in increasing order of complexity, and checking whether one of them implements the logic desired. Mission accomplished.
Surely this is not possible, as even quantum gates on itself already form an uncountable set. We will therefore restrict the quantum circuit space to a limited (finite) number of quantum gates, an alphabet say, on a case by case basis.
Brute Force Search on a Slice of Quantum Circuit Space
A little experimentation results in a fairly simple way to approach brute force search in this reduced context:
- Fix the number of wires to be used for the quantum circuit.
- Fix the "alphabet" of quantum gates to be used in the quantum circuit.
- Fix the maximum number of layers part of the quantum circuit.
- Generate all possible gate combinations for a single layer, using the gate alphabet and the number of wires.
- Scan all possible quantum circuits by trying all combinations of these layers, starting with quantum circuits consisting of a single layer, and incrementally moving up to quantum circuits consisting of the maximum number of layers.
To keep circuit composition simple, and as briefly mentioned before, we will represent controlled gates by a pair of single wire pseudo-gates: a generic control gate on the control wire and a wrapper for the controlled gate on the controlled wire. This simplifies the layer generation, but may yield invalid configurations, for example with only a control gate or multiple controlled gates. To resolve this, we will filter out invalid layers after generation.
As can be expected by now, Java code snippets are available at the end of this post, under the section An Implementation of Brute Force Search.
In what follows, we will provide a number of examples where we scan quantum circuit space for a circuit implementing a specific logic, using this brute force search implementation.
Wolfgang Pauli
As a first example of the brute force search technique, we will have a closer look at a collection of three gates defined by the matrices: $$X=\begin{bmatrix}0&1\\1&0\end{bmatrix},Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix}, Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}.$$ Here, \(X\) is the famous not gate.
These matrices are referred to as the Pauli matrices. They form a group of 16 elements, referred to as the Pauli group, that can be defined as $$G_1=\{fA,f\in\{\pm 1,\pm i\},A\in\{I,X,Y,Z\}\}$$ The Pauli group is generated by the elements \(X\), \(Y\) and \(Z\). The generation of the Pauli group based on these elements can be visualised using a Cayley graph:
Here, blue lines indicate right multiplication with \(X\), red lines indicate right multiplication with \(Y\) and green lines indicate right multiplication with \(Z\). As all three of the generators are their own inverse, the connections are bidirectional.
Note that all three generators are required to cover the entire group. Leaving one generator out covers only half of the group.
On the Cayley graph, you can easily see for example that \(XZ=-iY\). This can be verified using the brute force technique, with the Pauli gates as an alphabet, a matrix equality check with the matrix \(-iY\) and a maximum of 2 layers. The search finds the quantum circuit:
We verify the matrix multiplication by hand, just this one time $$XZ=\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}1&0\\0&-1\end{bmatrix}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}=-iY$$ Remember, for a series of single wire gates, the order of the gates is reversed for the matrix multiplication.
The hardest element of the Pauli group to generate with generators \(X\), \(Y\) and \(Z\), in terms of the number of operations, is \(-I\). We execute a brute force search, again with the Pauli gates as an alphabet, a matrix equality check with the matrix \(-I\), and a maximum of 4 layers. The search finds six quantum circuits, one of which is:
The other quantum circuits found, written in gate order, not matrix order, are \(XZXZ\), \(YXYX\), \(YZYZ\), \(ZXZX\) and \(ZYZY\). It is easy to verify on the Cayley graph that these are all the shortest paths to derive \(-I\), starting at \(I\), with length 4.
Jacques Hadamard
Extending on the reach of the Pauli gates is the Hadamard gate, represented as $$H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix}.$$ Using the alphabet of the Pauli gates, extended with the Hadamard gate, we will use brute force search to find the inverse of the Hadamard gate.
In order to achieve this with our scanner, we create a checker that takes a matrix and checks whether the output of a circuit is the inverse of that matrix, by multiplying both and comparing the result with \(I\).
Brute force search for the inverse of \(H\) with a maximum of 1 layer yields
This should not surprise us, as all quantum gates are represented by a unitary matrix, in particular the Hadamard gate, such that $$HH^\dagger=I.$$ And as \(H\) is its own conjugate transpose, it is also its own inverse.
We invite you to have a quick look at the Hadamard transform, to understand where the gate got its name.
Georges Remi
We will once more extend our reach and introduce the \(T\) gate and its conjugate transpose. These gates are defined as \(\pi/4\) rotations around the \(Z\) axis of the Bloch sphere: $$T=\begin{bmatrix}1&0\\0&e^{\frac{i\pi}{4}}\end{bmatrix}, T^\dagger=\begin{bmatrix}1&0\\0&e^{-\frac{i\pi}{4}}\end{bmatrix}.$$ We also introduce the phase gate \(S\), defined as a \(\pi/2\) rotation around the \(Z\) axis: $$S=\begin{bmatrix}1&0\\0&i\end{bmatrix}.$$ Clearly, given these definitions, \(S=TT\). But what's the inverse of S ?
Using the alphabet of all quantum gates that have been introduced so far, extended with \(T\), \(T^\dagger\) and \(S\), brute force search to find \(S^{-1}\), with a maximum of 2 layers yields
Besides this circuit, other solutions found are \(SZ\) and \(ZS\). Check it out.
Daniel Gottesman and William Clifford
Another interesting collection of gates is the Clifford group. In Maris Ozols' paper on the Clifford group, we read that the group is generated by \(H\) and \(S\), the latter referred to as \(P\) in the paper.
We would love to present a Cayley diagram for the Clifford group, at least the one for 2x2 matrices. However, strictly speaking the group is defined as the unitaries that normailze the Pauli group, or as a quotient group, so that will require some extra work. For now, we note that we can generate 192 different 2x2 matrices using \(H\) and \(S\) as generators.
The Pauli gates can be expressed in function of the generators of the Clifford group. So let's try that, generating \(X\), \(Y\) and \(Z\) using \(H\) and \(S\) as generators.
A brute force search over the alphabet \(H\) and \(S\), a matrix equality check with the matrix for \(Z\) and a maximum of 2 layers finds
A brute force search over the same alphabet, checking for the matrix for \(X\) requires 4 layers to find
Finally, checking for the matrix for \(Y\), we need 8 layers to find
John Bell
The Bell states are states of two qubits with maximal entanglement. The concept of the Bell states has been used by John Bell in 1964 to prove that the result of some quantum mechanical experiments cannot be explained by a theory of local hidden variables, although the premise of statistical independence leaves room for alternative views, as argued by superdeterminism.
We will refer to a quantum circuit that can produce the Bell states out of the computational basis states as a Bell circuit. And we denote that circuit using \(B\).
We will try to find a Bell circuit using brute force search, using the same alphabet as before. The checker will apply the quantum circuit to the first computational basis state for two qubits and verify that the output is the first Bell state: $$B|00\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}.$$ A search limited to two layers finds
The search also finds
As an illustration of the matrix representation rules, we will go through the trouble of verifying the first circuit in terms of matrix calculus and verify the circuit logic. The rules stipulate $${\oldstyle M}(B)={\oldstyle M}(C\parallel C(X)){\oldstyle M}(H\parallel I)=(P_0\otimes I+P_1\otimes X)(H\otimes I)$$ For the first layer, we have $$ H\otimes I=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix}\otimes\begin{bmatrix}1&0\\0&1\end{bmatrix} =\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&1&0\\0&1&0&1\\1&0&-1&0\\0&1&0&-1\end{bmatrix}. $$ For the second layer, we get the well known matrix for the controlled \(X\) gate $$ \begin{align} P_0\otimes I+P_1\otimes X &=\begin{bmatrix}1&0\\0&0\end{bmatrix}\otimes\begin{bmatrix}1&0\\0&1\end{bmatrix}+\begin{bmatrix}0&0\\0&1\end{bmatrix}\otimes\begin{bmatrix}0&1\\1&0\end{bmatrix}\\ &=\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}+\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix} =\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}. \end{align} $$ And the multiplication of both is $$ {\oldstyle M}(B) =\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&1&0\\0&1&0&1\\1&0&-1&0\\0&1&0&-1\end{bmatrix} =\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&1&0\\0&1&0&1\\0&1&0&-1\\1&0&-1&0\end{bmatrix}. $$ So finally we can verify the application of the circuit matrix to the first computational basis state as $$ {\oldstyle M}(B){\oldstyle M}(|00\rangle) ={\oldstyle M}(B)\begin{bmatrix}1\\0\\0\\0\end{bmatrix} =\frac{1}{\sqrt{2}}\begin{bmatrix}1\\0\\0\\1\end{bmatrix} =\frac{|00\rangle+|11\rangle}{\sqrt{2}}. $$
Daniel Greenberger, Michael Horne and Anton Zeilinger
We now continue our exercise by looking for the quantum states with maximal entanglement of three qubits. These are known as (the simplest case of) the Greenberger-Horne-Zeilinger (GHZ) states.
Using the same technique as for the first Bell state, we will search for a circuit \(C\) of 3 wires, that when applied to the first computational basis state, results in maximal entanglement: $$C|000\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle).$$ We get our first result when searching for a 3 layer circuit:
Searching for a circuit with 4 layers, another solution shows up:
Looking at the Bell circuit and the GHZ circuit, you see the Hadamard gate introduces superposition and the controlled not gate introduces entanglement.
Felix Bloch
On wikipedia, we read that the controlled X gate seems to work like a classical gate in the computational basis \({|0\rangle,|1\rangle}\), but that things are different in the Hadamard tranformed basis \({|+\rangle,|-\rangle}\). Viewed in this basis, the state of the second qubit remains unchanged, and the state of the first qubit is flipped, according to the state of the second bit.
We invite you to try visualising this on this Bloch sphere.
Let's try to reproduce this by searching for alternative circuits for the controlled X gate. This can be done using the matrix checker, with the matrix of the controlled X gate we've seen before: $${\oldstyle M}(C\parallel C(X))=P_0\otimes I+P_1\otimes X.$$ Because the controlled X gate is in our alphabet (in the form of two single wire pseudo-gates \(C\) and \(C(X)\)), a myriad of trivial solutions appear. However, our targetted circuit is spawned as well:
Following the same path, we search for a circuit implementing the controlled \(Z\) gate, with matrix: $${\oldstyle M}(C\parallel C(Z))=P_0\otimes I+P_1\otimes Z.$$ Limiting the search to 3 layers yields 6 circuits, one of which is
Once more, we search for a circuit implementing the controlled \(Y\) gate, with matrix: $${\oldstyle M}(C\parallel C(Y))=P_0\otimes I+P_1\otimes Y.$$ Limiting the search to 4 layers this time yields 65 circuits, one of which is
and another one of which is
You get a feeling of why \(T\) is a convenient gate in quantum circuit assembly.
Don Coppersmith
To continue, we will introduce the phase gates $$R_N=\begin{bmatrix}1&0\\0&e^\frac{2\pi i}{2^N}\end{bmatrix}.$$ Note that the phase gates are a generalisation of gates we introduced earlier, as \(S=R_2\) and \(T=R_3\).
We will now try to find a circuit for the Quantum Fourier transform (QFT) with 3 wires, which was invented by Don Coppersmith.
On the same Wikipedia page, from the QFT Example, we learn that the matrix for that circuit is referred to as \(F_8\), where $$F_8=\begin{bmatrix}\omega^{rc}, r,c=0...7\end{bmatrix},\omega=e^{\frac{\pi i}{4}}.$$ We execute a brute force search, this time with the alphabet reduced to \(I\), \(H\), the control pseudo-gate \(C\), the controlled phase gate of order 2 \(C(R_2)\) and the controlled phase gate or order 3 \(C(R_3)\). Using the matrix checker with \(F_8\), and limiting the search to 6 layers yields 32 circuits, one of which is
Obviously, playing with the minimal selection of gates can only be done because the circuit we are looking for is known beforehand. However, with an alphabet limited to 5 gates and 3 wires we have 125 possible layer configurations, which reduces after filtering for invalid combinations of the pseudo-gates to 32 layer configurations. Scanning circuit space for a circuit with 6 of these layers implies looking at over 1 billion circuits. More on that in the next section.
Tommaso Toffoli
We've finally come to our original objective: using brute force search on quantum circuit space to find a circuit for the Toffoli gate.
The Toffoli gate is a doubly controlled not gate, flipping the quantum state of the third wire only if the first two wires have quantum state \(|1\rangle\). It is represented by
The Toffoli gate is universal in classical reversible computing, in the sense that it can be used to implement any boolean function on a number of bits. In quantum computing however, that is not the case. But being able to implement the Toffoli gate using a quantum circuit implies that quantum computers are able to compute anything classical computers can compute.
A quantum circuit that implements the Toffoli gate is
Clearly, our objective to find a circuit for the Toffoli gate using brute force search on quantum circuit space was formulated heavily underestimating the combinatorial complexity of quantum circuit space.
Using the same reasoning as before, with an alphabet limited to 6 gates and 3 wires we have 216 possible layer configurations, which reduces after filtering for invalid combinations of the pseudo-gates to 32 (yes, 32 again) layer configurations. Scanning circuit space for a circuit with 12 of these layers implies looking at over 1 quintillion (10^18, or 1 billion billion) circuits. And that's knowing the minimal alphabet we have to use, because we already know the solution.
I guess we'll need a quantum computer to do that.
Conclusions
Our objective of scanning quantum circuit space for circuits that implement a specific logic has been extremely useful in getting familiar with the domain of quantum computing.
Reformulating all aspects of quantum circuit simulation to use the matrix representation of quantum entities and matrix operations gave us a uniform computational framework and was able to support our entire exercise.
And we got to know a lot of smart people.
As for finding a circuit for the Toffoli gate, we were beaten by the numbers. We did stumble upon a number of pathways to overcome the combinatorial complexity, and approach the problem using divide and conquer. However, that will require some more study and experimentation. Maybe in a future blog post ...
Java Implementation
For our implementation of the brute force search, written in Java (version 8), we need a limited number of building blocks: a library for matrix representation and manipulation over the complex domain, an abstract data type for the representation and simulation of quantum circuits, and finally an implementation of the brute force process.
We will provide representative code snippets for each of those building blocks in the next sections.
A Matrix Library over the Complex Domain
Matrix libraries in Java are plentiful. Even if you require matrices over the domain of the complex numbers, there are still a number of good options.
However, as we only need a limited set op basic operations on complex numbers and matrices over complex numbers, we took the matrix development into scope of the exercise. We will not elaborate on that implementation here, as this is a marginal topic for this post. We will limit ourselves to stating that the implementation features immutable classes
Complex and Matrix,
as well as a class MatrixUtils that provides some matrix utility functions.
You will find references to the matrix operations in the code snippets in the next sections. We assume class and function naming to be self-explaining.
Simulating a Quantum Circuit
The simulation of a quantum circuit starts with an abstract data type for the representation of quantum circuits.
The first quantum entity we will model is the quantum gate. The
Gate interface specifies that a quantum gate is represented by a symbol and has an associated matrix.
public interface Gate {
public String getSymbol();
public Matrix getMatrix();
}
Commonalities between all gate implementations will be grouped in a base class NamedGate.
public class NamedGate implements Gate {
private String symbol;
private Matrix matrix;
public NamedGate(String symbol, Matrix matrix) {
this.symbol = symbol;
this.matrix = matrix;
}
...
}
The identity gate, which was omnipresent behind the screens, is implemented by the class IdentityGate as an extension of NamedGate.
public class IdentityGate extends NamedGate {
public static final Matrix I = MatrixUtils.getIdentity(2);
public IdentityGate() {
super("I", I);
}
}
It uses MatrixUtils::getIdentity to create a 2x2 identity matrix.
Note that it keeps a static final reference to the identity matrix.
Sharing the matrix of the identity gate over all instances can be done because the matrix data type is immutable.
The implementation of the Hadamard gate provides another example.
public class HadamardGate extends NamedGate {
public static final Matrix H = new Matrix(new Double[][] { { 1d, 1d }, { 1d, -1d } }).smul(1d/Math.sqrt(2d));
public HadamardGate() {
super("H", H);
}
}
This time, the matrix is created inline, again using a static final reference.
Note that the Matrix::smul function implements the multiplication of a matrix with a scalar (Double) value.
As explained before, control gates are not implemented as 2-wire gates, but rather as a combination of two single wire pseudo-gates. The
ControlGate class implements the part on the control wire.
public class ControlGate extends NamedGate {
public ControlGate() {
super("C", null);
}
}
It merely acts as a placeholder, as you will see below.
The pseudo-gate for the target wire is implemented by the
ControlledGate class,
which acts as a wrapper for the controlled gate and in itself also implements the Gate interface.
public class ControlledGate extends NamedGate {
private Gate gate;
public ControlledGate(Gate gate) {
super("C("+gate.getSymbol()+")", gate.getMatrix());
this.gate = gate;
}
...
}
You will see these pseudo-gates in action in a moment.
Next comes the implementation of a layer of a quantum circuit. We opted for the representation of a circuit as a sequence of layers, rather than a sequence of wires, which might seem more natural. However, as explained in the section Matrix Representations, the matrix representing a quantum circuit is not constructed out of wire matrices, but rather out of layer matrices.
A layer basically is an ordered sequence of gates. It is implemented by the
Layer class.
public class Layer {
public static final Matrix I = new Matrix(new Double[][] { { 1d } } );
public static final Matrix K0 = new Matrix(new Double[][] { { 1d }, { 0d } } );
public static final Matrix K1 = new Matrix(new Double[][] { { 0d }, { 1d } } );
public static final Matrix P0 = K0.vmul(K0.transpose());
public static final Matrix P1 = K1.vmul(K1.transpose());
private Gate[] gates;
private Matrix matrix;
public Layer(Gate ... gates) {
this.gates = gates;
}
...
public Matrix getMatrix() {
if (matrix==null) {
if (!containsControl()) {
matrix = I;
for (int wire=0; wire<gates.length; ++wire) {
matrix = matrix.tmul(gates[wire].getMatrix());
}
} else {
Matrix m0 = I;
Matrix m1 = I;
for (int wire=0; wire<gates.length; ++wire) {
Gate gate = gates[wire];
if (gate instanceof ControlGate) {
m0 = m0.tmul(P0);
m1 = m1.tmul(P1);
} else if (gate instanceof ControlledGate) {
ControlledGate cgate = (ControlledGate)gate;
m0 = m0.tmul(IdentityGate,I);
m1 = m1.tmul(cgate.getControlledGate().getMatrix());
} else {
m0 = m0.tmul(gate.getMatrix());
m1 = m1.tmul(gate.getMatrix());
}
}
matrix = m0.add(m1);
}
}
return matrix;
}
...
}
It is initialized using a variable number of gates.
The code snippet shown here only adds the matrix calculation to that.
The matrix is calculated lazily, on first request, and kept to satisfy subsequent requests.
The matrix calculation takes a different implementation with and without a controlled gate, as explained in the section Matrix Representations. Noting that
Matrix::add implements matrix addition and Matrix::tmul implements the tensor product,
the code will speak for itself: calculating a multi-factor tensor product or the sum of two multi-factor tensor products, with a factor for each wire.
In both cases, we start off with the 1x1 identity matrix, a kind of neutral element for the tensor product. It simplifies the code for the loop and has the desired effect. Check it out.
Note that this implementation can only support a single controlled gate per layer. This has also been mentioned in the section Matrix Representations.
As for gate implementations, a number of utility matrices are created and kept as final static references. Here we observe
Matrix::vmul as the matrix (vector) product and Matrix::transpose to calculate the conjugate transpose of a matrix.
Finally, we can implement the
Circuit class.
public class Circuit {
private List<Layer> layers;
private Matrix matrix;
public Circuit(int wires) {
layers = new ArrayList<>();
matrix = MatrixUtils.getIdentity(1<<wires);
}
...
public Matrix getMatrix() {
return matrix;
}
public void addLayer(Layer layer) {
matrix = layer.getMatrix().vmul(matrix);
layers.add(layer);
}
...
}
The Circuit class works as an accumulator of layers, and therefore is not immutable.
That choice is made in function of our particular use case.
The matrix is therefore not calculated lazily, but incrementally, as the circuit grows.
The matrix of an empty circuit is the identity matrix. This is not only correct, but again simplifies the matrix calculation when a layer is added. Note that for \(N\) wires, the matrix dimensions are 2\(^N\)x2\(^N\).
And magically, the implementation of the
addLayer function ensures the order of the matrix multiplications falls into place, reversed.
An Implementation of Brute Force Search
As expained in the section Brute Force Search on a Slice of Quantum Circuit Space, the only building blocks we need are the generation of layers, as all possible combinations of gates, scanning all possible combination of these layers up to a maximum number of layers, and checking the outcome of a quantum circuit applied to a collection of qubits,
The
LayerGenerator class takes an alphabet of gates, and generates all possible combinations of these gates over a fixed number of wires.
public class LayerGenerator {
private int wires;
private List<Gate> gates;
private Layer currentLayer;
private List<Layer> generatedLayers;
public LayerGenerator(int wires, List<Gate> gates) {
this.wires = wires;
this.gates = gates;
// layer generation
generatedLayers = new ArrayList<>();
currentLayer = new Layer(wires);
generateLayers(0);
}
public List<Layer> getLayers() {
return generatedLayers;
}
private void generateLayers(int wire) {
if (wire==wires) {
if (currentLayer.validate()) {
generatedLayers.add(currentLayer.copy());
}
} else {
for (Gate gate : gates) {
currentLayer.setGate(wire, gate);
generateLayers(wire+1);
currentLayer.setGate(wire, null);
}
}
}
}
The currentLayer member variable accumulates gates in a layer.
The generatedLayers member variable accumulates generated layers as a list.
The
generateLayers function generates the layers, recursively, filling out currentLayer, wire by wire,
and spawning a layer when the number of wires is reached.
Note the validation of the layer at this point, to check whether the number of control and the number of controlled gates are consistent.
A copy of the layer is made as it is used destructively during the generation process. We know, these layers are not as immutable as we'd hoped they would be.
The
CircuitChecker interface captures the essence of a circuit checker:
give it the matrix of a quantum circuit and it will check whether that is the desired matrix, one way or another.
public interface CircuitChecker {
public boolean check(Matrix m);
}
Different implementations have been used for this post.
The simplest one is the MatrixCircuitChecker, which simply compares the matrix of the circuit with a target matrix.
public class MatrixCircuitChecker implements CircuitChecker {
private Matrix matrix;
public MatrixCircuitChecker(Matrix matrix) {
this.matrix = matrix;
}
@Override
public boolean check(Matrix m) {
return matrix.equals(m);
}
}
Other types of checkers used in this post are fairly simple to implement, so we will not be spending time on them.
This brings us to the work horse of the brute force search, the circuit finder, implemented by the
CircuitFinder class.
This class is responsible for scanning all combinations of layers.
public class CircuitFinder {
private List<Layer> layers;
private CircuitChecker checker;
public CircuitFinder(List<Layer> layers, CircuitChecker checker) {
this.layers = layers;
this.checker = checker;
}
public void find(int depth) {
find(MatrixUtils.identity(1<<layers.get(0).getWires()), new Stack<>(), depth);
}
private void find(Matrix matrix, Stack<Layer> stack, int depth) {
// stream and test
List<Matrix> matrices = layers.stream()
.map(layer -> check(matrix, stack, layer))
.collect(Collectors.toList());
if (depth>1) {
// iterate and recurse
for (int i=0; i<matrices.size(); ++i) {
stack.push(layers.get(i));
find(matrices.get(i), stack, depth-1);
stack.pop();
}
}
}
private Matrix check(Matrix matrix, Stack<Layer> stack, Layer layer) {
matrix = layer.getMatrix().vmul(matrix);
if (checker.check(matrix)) {
stack.push(layer);
// found a matching circuit, handle it
stack.pop();
}
return matrix;
}
}
The class is constructed with a list of possible layers, and a checker to verify the matrix of a candidate circuit.
The search technique used is a depth first search, up to a maximum depth. All nodes in the search tree are checked, not only the leaf nodes. The depth can be increased gradually, until hopefully a result is found in a reasonalbe time. As such, the search provides an implementation of iterative deepening depth first search.
Again, the search is implemented in a recursive way, accumulating layers on a stack. At each level in the recursion, all possible layers for the level are mapped to the matrix representation of the circuit generated so far, with that layer added. These matrices are checked. Subsequently, if the maximum depth has not been reached (counting down instead of up), all these matrices are handled by a recursive call.
Honesty requires us to admit that we actually do not use the
Circuit class here, as the layers are available on a stack, and the matrix has been calculated already, for efficiency reasons.
Dragging the matrix along while the stack of layers grows, saves us a myriad of matrix multiplications.
Finally, we can put all these building blocks together and orchestrate a brute force search. We will use the example of finding a quantum circuit for the controlled \(Z\) gate, as documented in the section on our honnorable friend Felix Bloch.
public class BruteForceSearch {
...
public void findControlledPauliZCircuit() {
findCircuit("C(Z)", getControlledPauliZMatrix(), 3);
}
public void findCircuit(String symbol, Matrix matrix, int depth) {
LayerGenerator generator = new LayerGenerator(2, getGateAlphabet());
CircuitChecker checker = new MatrixCircuitChecker(matrix);
CircuitFinder finder = new CircuitFinder(generator.getLayers(), checker);
finder.find(depth);
}
private Matrix getControlledPauliZMatrix() {
return Layer.P0.tmul(IdentityGate.I).add(Layer.P1.tmul(PauliZGate.Z));
}
private List<Gate> getGateAlphabet() {
List<Gate> gates = new ArrayList<>();
gates.add(new HadamardGate());
gates.add(new PauliXGate());
gates.add(new PauliYGate());
gates.add(new PauliZGate());
gates.add(new ControlGate());
gates.add(new ControlledGate(new PauliXGate()));
gates.add(new PhaseGate());
return gates;
}
}
The getGateAlphabet function returns the alphabet as a list of gates. Here we are considering \(H\), \(X\), \(Y\), \(Z\) and \(S\), and the speudo-gates \(C\) and \(C(X)\).
The
getControlledPauliZMatrix function returns the matrix for the circuit implementing the controlled \(Z\) gate, the target matrix of the search, equally documented in the section on Felix Bloch.
The actual search is implemented by the
findCircuit function, that instantiates a circuit generator for 2 wires over said alphabet, instantiates a matrix circuit checker using said matrix,
and instantiates a circuit finder with the layers generated by the circuit generator and the checker.
It then start the search.
All this orchestration is activated by the
findControlledPauliZCircuit function.
The result once more is documented in the section on Felix Bloch.
Conclusions
Simulating a quantum circuit in Java, in the narrow sense as explained in the section Brute Force Search on Quantum Circuit Space, is fairly straightforward to realize. It boils down to creating an abstract data type for quantum circuits and their constituents, and implementing the top-down recipe for calculating the matrix of the quantum entities.
The code base for the brute force search is limited in size, and fairly straightforward as well.
Areas of improvement are identified as:
- Introducing the uniform approach described in the section Matrix Representations to the abstract data type, using a single interface reflecting the commonalities of all quantum entities.
- Extending the quantum circuit abstract data type to support multiple controlled gates or gates with multiple control wires on a single layer.
- Unifying the generation of all combinations of gates in a layer and layers in a circuit, as both follow the "take N out of M with repetition" pattern.
- Introducing parallel processing for brute force search to exploit the multi-threading potential of the hardware used.
Impressive work! Being able to find quantum circuits for particular logic fast would boost the application of quantum computing. Next blog, how to do the brute force search with a quantum computer??
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