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.
117
questions
1
vote
3answers
80 views
Find unitary such that $U:|i\rangle|0\rangle\rightarrow|i\rangle|A_i\rangle$
Let's assume I have two qubits of state $|A_0\rangle$ and $|A_1\rangle$ correspondingly stored in a quantum memory. How do I find a Unitary $U$ that acts on another register of 2-qubits such that
$$U:|...
2
votes
2answers
47 views
How to implement $\sqrt{iSWAP}$ in Qiskit
I want to implement the $\sqrt{iSWAP}$ operator using simple operations in Qiskit such as it is done for the $iSWAP$ here or $\sqrt{SWAP}$ gate here. How can I do this? If possible I would like to ...
2
votes
1answer
37 views
How does MCPhaseGate/MCU1Gate works internally in qiskit?
I was curious about the implementation of MCPhase/MCU1Gate and how it works without ancilla qubits. I ended up checking the code of the some auxiliary (?) function ...
5
votes
1answer
162 views
More efficient implementation of $4$-qubit gate
While working on an error detection algorithm, I stumbled upon the problem of simplifying the following implementation
Here, the $S$ gate is defined by
$$S=\left(
\begin{array}{cc}
\frac{\sqrt{3}}{2}...
0
votes
0answers
16 views
Compiling Quantum Circuits using the Palindrome Transform
This paper shows a way to produce optimal circuits. I haver verified most of them and they are correct except this procedure:
procedure ProduceArray(n)
I cannot ...
3
votes
1answer
115 views
How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?
How can I construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?
I am stuck in this and any help would be appreciated.
2
votes
0answers
39 views
Representation of multiple qubit matrices in Dirac notation
Imagine one wants to represent the and function for any number of qubits in Dirac notation. The and gate flips the target qubit if all the control qubits are in state 1. This is its matrix ...
6
votes
3answers
81 views
How to create the state $\vert 0 \rangle+i \vert 1 \rangle$ using elementary gates?
I am trying to write $|0\rangle+i|1\rangle$ in terms of elementary gates like H, CNOT, Pauli Y, using the IBM QE circuit composer.
I was thinking some kind of combination of H and Y since $Y|0\rangle=...
3
votes
1answer
83 views
Gate-level implementation of Eigenvalue-Inversion in HHL
I am trying to understand how does the gate-level implementation of eigenvalue-inversion step in the HHL algorithm works.
I am following this reference, where it is stated (Lemma 4) that this can be ...
6
votes
3answers
91 views
Decomposing gates resembling exponentiated members of desired gateset
Suppose I have access to a pretty typical gate set, for example $\{\text{CNOT}, \text{SWAP}, \text{R}_{x}, \text{R}_{y}, \text{R}_{z}, \text{CR}_x, \text{CR}_y, \text{CR}_z\}$ where $\text{CR}$ is a ...
-1
votes
1answer
30 views
How to create a gate with functionality CCX(a,b,b)?
Can we create a Controlled gate with below functionality?
if {a==|1> && b==|1>} then {qc.x(b)}
Basically, a CCX gate but the output Qubit is actually one of the input Qubits. Apparently, ...
5
votes
0answers
43 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 ...
4
votes
1answer
126 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
150 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
97 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)):
$$
U=\pmatrix{
0 &...
1
vote
1answer
56 views
Principal square root of Pauli Y gate in Qiskit?
I've seen a similar question asked (How do I compute the square root of the $Y$ gate?) but I'm trying to understand how I can use the gates $Y^{\frac{1}{2}}$ or $Y^{\frac{1}{4}}$ in Qiskit in terms of ...
2
votes
1answer
86 views
How to define Q-operator in Quantum Amplitude Estimation
I'm trying to implement a circuit for Quantum Amplitude Estimation in Qiskit using elementary gates.
I have created the circuit that represent my algorithm $A$ but now from the theory I know that I ...
4
votes
1answer
58 views
Adding a phase to qubit: why is it necessary for arbitrary single qubit gate
An arbitrary single qubit gate can be decomposed as:
$$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 ...
4
votes
1answer
135 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 ...
5
votes
0answers
38 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
152 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
1answer
177 views
Is it possible to express $U_1(\lambda)$ through the gates $R_x, R_y, R_z$ while maintaining the phase? In Qiskit for example
Is it possible to express gate $U_1(\lambda)$ through the gates $R_x, R_y, R_z$ while maintaining the phase? Both in principle and in practice (in Qiskit for example)?
The single gate $R_z(\lambda)$ ...
0
votes
0answers
32 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
103 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 ...
2
votes
2answers
117 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
89 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
140 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
67 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
68 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
42 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$ ...
2
votes
0answers
61 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 ...
5
votes
2answers
312 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
102 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
46 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 ...
9
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
91 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:
$$
U_f = \left(\begin{array}{cccccccccc}
1 & 0 & 0 & \dots & \dots & \dots & \dots & \dots & \dots &...
6
votes
1answer
108 views
How to factor Ising YY coupling gate into product of basic gates?
Let us consider Pauli YY coupling gate of the following form
$$
YY_\phi=
\left(\begin{matrix}
\cos(\phi) & 0 & 0 & i \sin(\phi) \\
0 & \cos(\phi) & -i \sin(\phi) & 0 \\
...
2
votes
1answer
537 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
698 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
151 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
86 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 ...
10
votes
1answer
166 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
146 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
78 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
143 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
295 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
54 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
310 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
442 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.
$$\...
