Questions tagged [gate-synthesis]
For questions about finding (short) gate sequences to implement a specific unitary operation, for example decomposing a complicated multi-qubit gate into a sequence of basic gates. It might apply to optimizing circuits with respect to length or depth or finding gate sequences to implement an algorithm.
87
questions
1
vote
0answers
20 views
Decompose Toffoli gate with minimum cost for IBM quantum computer
The known decomposition of toffoli gate that can be used on IBM quantum computer is :
I want to know any other Toffoli gate decompositions that can be used on IBM quantum computer and have a cost ...
3
votes
2answers
191 views
How can I fill a unitary knowing only its first column?
I have a unitary matrix that I want to construct. I only care what happens to the first computational state, so the first column is specified. So far, I've been assigning each question mark to a ...
2
votes
2answers
78 views
How to prove that a matrix is an arbitrary unitary?
My goal is to prove that I can synthesise arbitrary unitary from two components.
In the end, I find a matrix with the form
\begin{equation}
\mathbf{W}_j=\begin{pmatrix}
|\alpha|2\cos{(\phi_{...
3
votes
0answers
33 views
Cost of controlled-$U_i$
What is the cost (number of gates) of $\sum_{i=0}^{N-1}| i \rangle \langle i|\otimes U_i$ in terms of $N$ and the costs of the unitaries $U_i$? Say the gate set consists of arbitrary one-qubit gates ...
8
votes
3answers
1k views
How can I see, without math, the action of a gate in matrix form?
Suppose we have the Fredkin gate with
$$
F=
\left( {\begin{array}{cc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 ...
4
votes
2answers
72 views
How to construct a circuit to perform this operation? Is there a general way of getting a circuit from a matrix?
I want to build a circuit that performs the following operation:
$$
U_f = \left(\begin{array}{cccccccccc}
1 & 0 & 0 & \dots & \dots & \dots & \dots & \dots & \dots &...
6
votes
1answer
58 views
How to factor Ising YY coupling gate into product of basic gates?
Let us consider Pauli YY coupling gate of the following form
$$
YY_\phi=
\left(\begin{matrix}
\cos(\phi) & 0 & 0 & i \sin(\phi) \\
0 & \cos(\phi) & -i \sin(\phi) & 0 \\
...
2
votes
1answer
262 views
Decomposing a controlled phase gate into CNOTs
I'm trying to understand the following derivation of decomposing a controlled $R_k$ (phase) gate into a combination of CNOTs and single qubit gates, but there's one main thing about the process that ...
0
votes
0answers
28 views
Help with understanding Unitary Operator of Quantum Optical Fredkin Gate
I'm referring to the textbook "Quantum Computation and Quantum Information" 10th Anniversary Edition by Nielsen and Chuang. Chapter 7.4 has a Box 7.4 which introduces the Quantum Optical Fredkin Gate....
1
vote
0answers
438 views
How to implement controlled u3 gate from Qiskit using simpler gates?
I am trying to implement the u3 controlled gate (able to rotate the qubit in any specified direction in 3 dimensions if the control is 1, for two qubits) using simpler gates. The simpler gates ...
3
votes
1answer
109 views
control gate with 3 inputs, two control and rotation gate
My question is about if there is any way to represent a circuit that take 3 inputs and applies a rotation gate on the third qubit if the first two qubits is similar (has the same state)?
1
vote
1answer
48 views
Efficient implementation of exponential of projector
If I have an $n$ qubit system and a projector $P$ such as $P_0 = \left|0\right>^{\otimes n}\left<0\right|^{\otimes n}$ (as an example) on those qubits, is there an efficient way to implement the ...
9
votes
1answer
106 views
Sampling random circuits vs Solovay-Kitaev compiler
Suppose I want to obtain a gate sequence representing a particular 1 qubit unitary matrix.
The gate set is represented by a discrete universal set, e.g. Clifford+T gates or $\{T,H\}$ gates.
A well ...
3
votes
2answers
91 views
Calculate the square root of Euler angles
I am trying to find a nice way to represent the square root of an arbitrary single qubit unitary to implement Lemma 6.1 from this paper
Given the Euler angles: $R_z(a)R_y(b)R_z(c) = \left(R_z\left(a'\...
1
vote
1answer
48 views
Approximating unitary matrices — restricted gateset
Note: This question is a follow up of Approximating unitary matrices.
The decompositions provided in Approximating unitary matrices are correct and worked for me without problem.
But I am now facing ...
3
votes
1answer
98 views
How to create an $n$-qubit normally controlled gate?
Suppose I have a quantum gate $U$ and it's a controlled gate. In particular, I have a $2\times 2$ matrix formulation of the gate's action on 2 adjacent qubits.
How can I make this work on an $n$-bit ...
2
votes
1answer
150 views
Using Quantum Fourier Transform in adding two 2-bit numbers
I am trying to use Qiskit to write a code that uses QFT to add 2 numbers. I am referring to this paper:
https://iopscience.iop.org/article/10.1088/1742-6596/735/1/012083
I have a few questions:
1) Is ...
1
vote
0answers
35 views
Controlled controlled adder gates involved
Let's say I have a circuit that given in the figure
As we can see that this circuit consists of $2$-Toffoli gates and $4$ C-NOT gates, and to construct this entire circuit using only single qubit ...
6
votes
1answer
166 views
How to decompose a controlled unitary $C(U)$ operation where $U$ is a 2-qubit gate?
In the vein of this question, say I have a 2-qubit unitary gate $U$ which can be represented as a finite sequence of (say) single qubit gates, CNOTs, SWAPs, cXs, cYs and cZs. Now I need to implement a ...
5
votes
2answers
277 views
Transform matrix into a combination of simple quantum gates
I am trying to transform this matrix into a combination of quantum gates but I cannot find any such functionality on Qiskit or anywhere else. I have tried to use Quirk but I do not understand it.
$$\...
5
votes
1answer
143 views
Nielsen and Chuang's proof for 'approximating arbitrary unitary gates is generically hard'
The following statement is found on the page 199 of Nielsen and Chuang's book (10th Anniversary Edition) in the proof for the fact that 'approximating arbitrary unitary gates is generically hard':
...
1
vote
0answers
90 views
Transferring classical OR gate in a quantum gate
I would be interested to know how to transform the classic OR gate into a quantum gate. I thought a little about myself. The OR gate can also be rewritten as a NAND gate:
So, I have now tried to ...
3
votes
2answers
120 views
Minimal quantum OR circuit
The quantum OR circuit between $|a\rangle$ and $|b\rangle$ can be made out of 1 Toffoli and 2 CNOT gates, 1 ancillary qubit. Is there any other implementation? Or is this the minimal in the sense of ...
3
votes
1answer
75 views
Making a maximally mixed 2-qubit state in the IBM Q
I am trying to make a 2-qubit maximally mixed state $\mathbb{I}/4$ where $\mathbb{I}$ is the identity $4\times 4$ matrix.
I know that, for a maximally mixed 1-qubit state I can use a Hadamard gate, ...
8
votes
2answers
379 views
How is it possible to implement unitary operator when its size is exponential in inputs?
A quantum circuit can use any unitary operator. Its matrix is exponential in the number of input bits. In practice how can this ever be possible (aside from operators which are tensor products), i.e. ...
2
votes
1answer
111 views
Implementing these $N×N$ matrices on $\log N$ qubits
Consider $n$ qubits and the $N=2^n$ states that I label
\begin{equation}
|k \rangle = \sum_{i=0}^{n-1} 2^i q_i,
\end{equation}
i.e. $|q_{n-1}\cdots q_0 \rangle \rightarrow |k\rangle$, where $q_j \in \...
0
votes
0answers
33 views
How are multi-qubit gates extended into larger registers? [duplicate]
Implementing a single-qubit gate in a multi-qubit register is relatively easy. For example, this gate:
This is equivalent to $I \otimes H \otimes I$. If the $H$ gate was on the first bit, it would be ...
5
votes
2answers
300 views
Implement Fredkin gate with square root of swap
I would like to implement a Fredkin gate based on square root of swap and one-qubit gates. In particular, I was hoping to find the exact gate named "?" in this circuit:
In addition, I want to avoid ...
4
votes
1answer
99 views
Rewrite circuit with measurements with unitaries
In quantum physics, because of the no-cloning theorem, lots of classical proofs of cryptographic problems cannot be turned into quantum proofs (rewinding is usually not possible quantumly). A dream ...
2
votes
1answer
71 views
Optimise implementation of a quantum algorithm when an input is fixed
I need to implement a quantum comparator that, given a quantum register $a$ and a real number $b$ known at generation time (i.e. when the quantum circuit is generated), set a qubit $r$ to the boolean ...
2
votes
1answer
301 views
Building an N-qubit Controlled S Gate
I've been beating my head against this problem for three days now and I just can't seem to crack it. To construct an N-qubit controlled Unitary gate, I can do something like this (note I'm using ...
2
votes
1answer
119 views
If CNOTs and single qubit gates are universal then why do we need to prove that controlled U operations can be composed by them as well?
In the book by Chuang and Nielsen they prove that controlled U operations can be made out of CNOTs and single qubit gates. But then they go on to prove that they are universal by showing that every n ...
2
votes
1answer
422 views
Decompose a general two-qubit gate into general controlled-qubit gates
We often seek to decompose multi-qubit unitaries into single-qubit rotations and controlled-rotations, minimising the latter or restricting to gates like CNOTs.
I'm interested in expressing a general ...
2
votes
1answer
120 views
How to implement a $\frac{\theta}{2}$ rotation from $\theta$ rotation?
Is there a way to create a rotation gate which has half the angle of some implementable gate?
I am looking to implement a gate on Quirk which allows for standard time-dependent rotations
$$R_x(\...
3
votes
0answers
122 views
Decomposition of any 2-level matrix into single qubit and CNOT gates
I saw an example which takes a 2 level matrix. Which is a $8\times8$ matrix that acts non trivially only on 2 levels of only states $|000\rangle$ and $|111\rangle$. The way they do it is by using a ...
4
votes
2answers
1k views
Composing the CNOT gate as a tensor product of two level matrices
I don't understand, why is the control not gate used so often. As far as I understand it, if you apply two 2 level operations on two qubits then you get a 4 x 4 matrix by the tensor product. So how ...
1
vote
1answer
83 views
Proof that $2^n \times 2^n$ operator be decomposed in terms of $2 \times 2$ operators
What is the proof that any $2^n\times 2^n$ quantum operator can be expressed in terms of the tensor product of $n$ number of $2\times 2$ quantum operators acting on a single qubit space each?
6
votes
3answers
662 views
Is there any method of adding two operators in a circuit?
I am trying to reconstruct the time evolution of a Hamiltonian on the quantum computing simulator, quirk. Ideally I would like to generalise this to any simulator. The unitary matrix is
$$U(t)=e^{-...
4
votes
3answers
247 views
Hadamard gate as a product of $R_x$, $R_z$ and a phase
I am having problems with this task.
Since the Hadamard gate rotates a state $180Ā°$ about the $\hat{n} = \frac{\hat{x} + \hat{z}}{\sqrt{2}}$ axis, I imagine the solution can be found the following ...
1
vote
1answer
76 views
Creating a time dependent custom gate in Quirk
I have created a $16\times 16$ unitary operator using a Hamiltonian by finding its exponential
$$U=\exp(-iH\delta t)$$
If I set $\delta t=1$ then I can take this matrix and input it into quirk using ...
1
vote
1answer
264 views
Gate sequence for exponential of product of Pauli Z operators
I want to compile $$\exp(-i \theta \sigma_i^z \sigma_j^z)$$ down to a gate sequence of single qubit rotations and CNOTs. How do I do this? What is the general procedure for compiling a unitary $U$ to ...
10
votes
3answers
877 views
Approximating unitary matrices
I currently have 2 unitary matrices that I want to approximate to a good precision with the fewer quantum gates possible.
In my case the two matrices are:
The square root of NOT gate (up to a global ...
6
votes
5answers
2k views
How do I build a gate from a matrix on Qiskit?
I'm creating a gate for a project and need to test if it has the same results as the original circuit in a simulator, how do I build this gate on Qiskit? It's a 3 qubit gate, 8x8 matrix:
$$
\frac{1}{...
4
votes
1answer
252 views
Construction of ${R_n(\theta)}$ using only the Hadamard and ${\pi/8}$ gates
In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen & Chuang", they claim that Eqn(4.75) is a rotation about the axis along the direction
( ${cos(\pi/8)}$, ${...
3
votes
2answers
278 views
Implementing gate with two parameters using Qiskit in Python
I am trying to implement the HHL algorithm (for solving $Ax=b$). I am assuming $A$ to be unitary and Hermitian so that I can find the Hamiltonian simulation for it easily.
For any $A$ to be Hermitian ...
6
votes
2answers
680 views
How to obtain Y rotation with only X and Z rotations gates?
Let's say you have a system with which you can perform arbitrary rotations around the X and Z axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the Y axis?
...
5
votes
2answers
450 views
Measuring the Hamiltonian in the VQE
I am trying to implement VQE in pyQuil and am dumbfounded by how to measure the expectation value of a general Hamiltonian on $\mathbb{C}^{2^n}$ i.e. determine $\langle\psi , H \psi\rangle$ on a ...
5
votes
2answers
100 views
Clarification needed: “Simulation” of $e^{-iHt}$ and its time complexity
On page 3 here it is mentioned that:
However, building on prior works [32, 36, 38] recently it has been
shown in [39] that to simulate $e^{āiHt}$ for an $s$-sparse Hamiltonian
requires only $\...
3
votes
0answers
76 views
Circuit to construct a $n$-qubit state which is a superposition of states with only a single qubit being $\lvert1\rangle$ [duplicate]
So the question came up in a book I am working through. Given a circuit with $n$ qubits, construct a state with only $n$ possible measurement results, each of which has only $1$ of $n$ qubits as $1$, ...
2
votes
1answer
134 views
How to construct a quantum gate producing 1 if r divides x, 0 otherwise?
If you have two registers in the state $\frac{1}{2^{n/2}} \sum_{x = 0}^{2^{n/2} - 1} |x\rangle |0\rangle$, how could you construct a gate that produces a superposition of states $|x\rangle|1\rangle$ ...
