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.
105
questions
3
votes
0answers
15 views
Reducing the depth of quantum circuits with ancilla qubits
This question is two-fold and considers general $n$-qubit operations on a quantum computer.
First, can a general $n$-qubit operation be implemented on a quantum computer without the use of ancilla ...
2
votes
1answer
32 views
CNOT expressed with CZ and H gates by taking into account HZH =X
From this link:
Where equation 1 is:
I can probably brute-force this by explicitly calculating this quantum circuit's effective 4x4 matrix and seeing that its equivalent to this teleportation ...
0
votes
1answer
45 views
How to code a projector operator in qiskit?
I'm new to qiskit and I want to know how do I define a projector operator in qiskit? Specifically, I have prepared a 3 qubit system, and after applying a whole lot of gates and measuring it in a state ...
3
votes
2answers
65 views
Decomposition of $|110\rangle \leftrightarrow |000\rangle$ Exchange Gate
How to implement a 3 qubit gate, that exchanges the level $|110\rangle$ and $|000\rangle$, with elementary gates (CNOT, SWAP, Toffoli, local gates, etc.(everything Qiskit allows)):
$$
U=\pmatrix{
0 &...
1
vote
1answer
26 views
Principal square root of Pauli Y gate in Qiskit?
I've seen a similar question asked (How do I compute the square root of the $Y$ gate?) but I'm trying to understand how I can use the gates $Y^{\frac{1}{2}}$ or $Y^{\frac{1}{4}}$ in Qiskit in terms of ...
2
votes
1answer
30 views
How to define Q-operator in Quantum Amplitude Estimation
I'm trying to implement a circuit for Quantum Amplitude Estimation in Qiskit using elementary gates.
I have created the circuit that represent my algorithm $A$ but now from the theory I know that I ...
4
votes
1answer
39 views
Adding a phase to qubit: why is it necessary for arbitrary single qubit gate
An arbitrary single qubit gate can be decomposed as:
$$U=e^{i \alpha} R_z(\beta) R_y(\gamma) R_z(\delta)$$
We notice that in addition to the three rotations, there is a coefficient $e^{i \alpha}$. ...
7
votes
1answer
1k views
Would IBM's “compiler” turn my identity circuit into nothing?
If I were to create a circuit with the following gate:
$$\tag{1}R_\phi = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \phi} \end{bmatrix},$$
with $\phi$ specified to be equal to 0, then the gate that I ...
2
votes
1answer
41 views
How can you decompose Grover's diffusion operator into gates?
I know how Grover's diffusion operator works ($U_s = 2|s\rangle\langle s|-I$) with the inversion around the mean. However, I want to implement it in simpler gates, to use the algorithm. How can I do ...
4
votes
0answers
32 views
Equivalence checking of quantum circuits up to error
Suppose you are given two circuit descriptions $A$ and $B$ where by a circuit description I mean a sequence of gates (in the order they are applied) and the qubits they are applied on. (For the sake ...
4
votes
2answers
128 views
How to find a circuit for the roots of QFT?
After reading about using quantum gates instead of ancillas, it asserts that every quantum circuit has a square root. Theoretically, they do, but is there a practical method to generate the quantum ...
0
votes
0answers
30 views
[CNOT GATE]how to go from a passage matrix acting on |C0>,C1>|T0>,T1> to the Cnot matrix acting on |C0T0>,C0T1>
In article A controlled-NOT gate for frequency-bin qubits,
the authors built a passage matrix acting on the states $T_0$, $T_1$, $C_0$, $C_1$ and then they infered the famous two qubit matrix notation....
3
votes
2answers
72 views
Are Toffoli gates actually used in designing quantum circuits?
In an actual quantum computer, are we designing circuits with Toffoli Gates and then using compilers or optimizers to remove redundancies so that we can use fewer qubits than a full Toffoli gates ...
1
vote
2answers
82 views
How to construct a CU3 gate using only CX and U3 gates?
Knowing that CX and U3 (taking 3 parameters $\theta, \phi$ and $\lambda$) form a set of universal gates how can I construct an arbitrary CU3 gate using a decomposition of only CX and arbitrary U3 ...
2
votes
1answer
64 views
What are the differences between the Toffoli and Fredkin gates (historical, practical, etc.)
I'm trying to understand the historical ordering and the practical differences between the Toffoli Gate and the Fredkin Gate.
Toffoli's February 1980 tech report MIT/LCS/TM-151 states:
Where ...
3
votes
3answers
121 views
What is the complexity of splitting a state into a superposition of $n$ computational basis states?
$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$
I'...
3
votes
1answer
49 views
Implementing a controlled-controlled-U using controlled-U
Suppose I know how to implement a 2 qubit gate $C-U$ (i.e controlled U), and I want to implement $CC-U$ using $C-U$ and other 1 or 2 qubit gates, is that possible?
1
vote
1answer
57 views
Cheap Toffoli gates with phase errors
Here, a cheap verion of a Toffoli, up to a phase flip for $|101\rangle$, is given by
with $A=R_y(\pi/4)$. Are there similar versions of cheap implementation of general $C^nNOT$ gates?
I tried to ...
1
vote
0answers
41 views
How can I prove the universality of this set of gates?
I was reading this article. A brief explanation: Here we have a circuit, the registers are a stepfunction state, an single photon state and a function state. The first two have position operators $X$ ...
1
vote
0answers
36 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 ...
4
votes
2answers
257 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
94 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
42 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
83 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:
$$
U_f = \left(\begin{array}{cccccccccc}
1 & 0 & 0 & \dots & \dots & \dots & \dots & \dots & \dots &...
6
votes
1answer
72 views
How to factor Ising YY coupling gate into product of basic gates?
Let us consider Pauli YY coupling gate of the following form
$$
YY_\phi=
\left(\begin{matrix}
\cos(\phi) & 0 & 0 & i \sin(\phi) \\
0 & \cos(\phi) & -i \sin(\phi) & 0 \\
...
2
votes
1answer
423 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 ...
1
vote
0answers
636 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
125 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
76 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
134 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
124 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
66 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
119 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
216 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
45 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
245 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
365 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.
$$\...
7
votes
1answer
163 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
114 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
130 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
94 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
417 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. ...
3
votes
1answer
135 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
34 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
415 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
103 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
79 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
381 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
150 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 ...
