All Questions
Tagged with gate-synthesis quantum-gate
133 questions
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Procedure for constructing magic state gate injection gadgets
I have a quantum computer that cannot implement the gate $U$ by itself, and needs help in the form of a magic state $|U\rangle$ held in another register. Is there a general-purpose procedure for ...
1
vote
1
answer
255
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Simple successor gate
I want to create a gate that, given a sequence of qbits that encode n, transforms that sequence in n+1, in other words the successor function. I managed to do it in qiskit by writing this:
...
- 113
0
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0
answers
39
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4- controlled- Z gate (C4Z gate) formation using C4X and Hadamard gates
I require a quantum circuit that utilizes C4X and Hadamard gates to form a C4X gate. Could anyone please help me?
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3
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1
answer
155
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Is Controlled$(R_z(\theta))$ more expensive than Controlled$(Z^t)$ on the surface code?
There are (at least) two conventions for single-qubit, arbitrary-angle Z rotations in quantum computing, which I will call Rz(theta) and Z^t.
$$
R_Z(\theta) = \exp(-i \theta Z/2) = \mathrm{diag}(e^{-i ...
0
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1
answer
158
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Is there a tool to decompose 4-Qubit unitaries (aka 16x16 matrices)?
I was wondering if there is a tool that can decompose such a matrix in gates on 4 qubits?
I found one for 3-qubit gates (9x9 matrices) in Cirq but nothing for bigger matrices.
(The matrix is not ...
- 105
1
vote
1
answer
77
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Implementation of a unitary operator scaled by a factor
Is it possible to implement a unitary operator scaled by a factor on a quantum computer?
Let's say the unitary operator is $U$:
$$U=\begin{bmatrix}
u_0 & u_1 \\ u_2 & u_3
\end{bmatrix}\...
- 321
0
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1
answer
64
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Realization of the gate $(I\pm U)/2$
The state after applying the Hadamard test (before measurement)
is $$\newcommand{\ket}[1]{|#1\rangle}\newcommand{\bra}[1]{\langle#1|}\ket{0}\frac{I+U}{2}\ket{\psi} + \ket{1}\frac{I-U}{2}\ket{\psi}.$$
...
- 321
2
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2
answers
90
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Better constant for linear depth incrementers
Currently working on some quantum arithmetic and was wondering if we have a better constant factor for a linear depth incrementer.
As an example (and the best I could currently find), Craig Gidney ...
- 156
5
votes
4
answers
121
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$U_1\oplus U_2$ decomposable into $I\oplus U$ and 1-qubit gates?
TL;DR
Let $U_1, U_2, U$ be arbitrary 1-qubit quantum gates.
Can 2-qubit gates of the form $U_1\oplus U_2$ always be decomposed into a combination of controlled gates ($I\oplus U$) and single qubit ...
- 321
1
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1
answer
181
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How many gates are necessary to implement an arbitrary n-qubit permutation unitary?
How many gates are necessary to implement an arbitrary n-qubit permutation unitary, using only 1- and 2-qubit gates?
An n-qubit permutation unitary is a $2^n$ x $2^n$ unitary matrix where each entry ...
- 181
3
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1
answer
115
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Is it possible to decompose a controlled gate with control qubit in the $|+\rangle$ state?
$\newcommand{\ket}[1]{\vert#1\rangle}\newcommand{\bra}[1]{\langle#1\vert}$
Given a quantum circuit with 2 qubits that executes a controlled gate $CU$ where the control qubit is in the $\ket{+}$ state, ...
- 321
1
vote
0
answers
40
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Calculation of feasible operations for a certain set of primitive gates
Assume we have a set of primitive operations of a quantum processor. How can I determine the set of feasible operations or prove that a certain operation is not feasible?
As an example, one could ...
- 23
2
votes
1
answer
469
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Multi-control multi-target gate
I'm using qiskit for simulation. Suppose, I have 6 qubits with indices [0, 1, 2, 3, 4, 5] and I have an operator $U$ of size $4 \times 4$, such that it operates on ...
2
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1
answer
77
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Math Behind $X$ Gate With Arbitrary Phase is equivalent to $ZXZ$ Gate
An X gate where there is a phase shift $\phi$ to the applied sinusoidal wave $U = e^{-i\frac{\theta}{2}(cos(\phi)\sigma_x+sin(\phi)\sigma_y)}$ is equivalent to a series of gates $Z_{-\phi}X_{\theta}Z_{...
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0
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83
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Application of transformation $U_d$ that maps any qudit state to $|d-1\rangle$
When giving examples of universal gate sets in the paper Qudits and High-Dimensional Quantum Computing, the authors first define the transformation that maps any given qudit state to $|d-1\rangle$:
$$
...
- 867
2
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1
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225
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Exact synthesis of Toffoli gate from CNOT and rational single-qubit gates?
Is it possible to implement a Toffoli gate exactly using just CNOT gates and single qubit complex rational gates (i.e. with entries in $\mathbb{Q}(i)$), possibly with ancillas?
I know this works with ...
- 21
3
votes
1
answer
232
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How many two-qubit controlled gates do you need to simulate any CU gates where U is a diagonal matrix?
Assuming we have n qubit, the first qubit is a control qubit, and the rest are the targets of $U$.
If $U$ is a diagonal matrix, is there any theory to find the minimum number of two-qubit controlled ...
- 103
2
votes
1
answer
104
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Converting a Matrix to a Gate in OpenQasm 2
I am a beginner when it comes to quantum computing so forgive me if this is a dumb question. Does anyone know how to create a gate from any matrix on OpenQasm2? Specifically, can anyone provide any ...
- 31
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0
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85
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How to Trotterize a CNOT gate?
I came across a paper that said that they Trotterized a CNOT gate into 4 blocks of CU gates where the CU gate parameters are specified. This was all done on Qiskit. How does this Trotterization ...
- 51
1
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2
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1k
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How to construct common classical gates with CNOT circuit?
How can I construct AND, OR, NAND, NOR with CNOT gates.
First off, this other question describes how to make them with matrices.
Theoretically I can construct all those gates already. I know how to ...
- 113
5
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3
answers
858
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Quasiprobability decomposition of the CZ-gate
I was trying to obtain the quasi-probability decomposition of the CNOT gate by using the information in this paper.
The authors give us the example for the CZ gate (Figure 2, i.e. the one below).
The ...
- 513
0
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1
answer
123
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Implementing Odd Permutations Without Ancilla Bit
The paper says that
The inversion $\alpha \mapsto \alpha^{-1} $ (where 0 is mapped to 0)
can be seen as a permutation on $\mathbb F_{256}$. This permutation is odd, while
quantum circuits with NOT, ...
- 1
1
vote
1
answer
306
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Decomposition of unitary operator into rotations around Bloch sphere
I apologize in advance for any mistakes as I am new to this field and come from a programming, rather than mathematical/physical background.
I am looking for a way to decompose a given operator $U$ ...
1
vote
0
answers
67
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Faithful description of a photonic setting with the circuit model
The above picture comes from this paper.
The circuit on the left and the one on the right are equivalent (up to the basis).
However, there is an important difference: the circuit makes the input -- i....
- 2,058
2
votes
1
answer
126
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Is there a name for a gate that 'moves' one qubit to a new position via multiple SWAP gates?
Let's say there is a qubit at position $i$, and I want to move it to position $i'$. Without loss of generality, let's say $i < i'$. By 'move it' I mean, perform multiple $SWAP$ operations so that ...
- 1,459
1
vote
2
answers
159
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CNOT chain vs CNOT fountain in qiskit
I was going through qiskit's synthesis module, their methods take an argument called cx_structure which has two possible values, ...
- 391
4
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0
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100
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What is the correct name of this quantum gate? Possibly state control gate
Let $\vec v \in \mathbb{C}^2 $ be the following quantum state:
$$
\vec v = \frac{1}{\sqrt{2}}\begin{bmatrix}
v_{1} \\
v_{2} \\
\end{bmatrix},\space \lvert v_1 \rvert = 1,...
- 41
5
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1
answer
344
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Native Gate Decomposition
TL;DR: I've got a very small set of gates to use and need to find efficient decompositions for $R_y$ and controlled $R_y$ gates. Does anyone have any better ideas than what I have?
I'm looking to ...
- 474
4
votes
1
answer
159
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Confusion with the number of CNOTs in a circuit
I am a bit puzzled on the following circuit. According to this Quantum Computing SE thread it holds that
$$
e^{i(Z\otimes Z)t} = {\rm CNOT} (I\otimes e^{iZt}){\rm CNOT} \qquad (1)
$$
As a result we ...
- 83
6
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2
answers
724
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How many quantum gates are needed to prepare an arbitrary state?
In this paper there is this sentence:
[...] the description of a $2^n\times2^n$ unitary matrix $U$ (which is a poly($n$)-size quantum circuit)
According to the meaning of "which" in ...
- 481
2
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1
answer
85
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Decompose bell measurement gate into combination of controlled-not gates and one-qubit gates
OPENQASM2.0 has only one two-qubit gate: controlled not. For a teleportation experiment, I need to perform a measurement in the Bell basis. That is, I need a two-qubit gate with matrix representation
$...
- 701
1
vote
2
answers
113
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Circuit including phase factor in $XY(\beta, \theta)$ gate
In Implementation of the XY interaction family with calibration of a single pulse, the $XY(\beta, \theta)$ gate is defined as
$$
XY(\beta, \theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 &...
- 2,965
-1
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1
answer
130
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Decomposition of U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate [closed]
Can we express U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate
$$ U_1(\lambda) = \begin{pmatrix}1 & 0 \\ 0 & e^{i\lambda}\end{pmatrix}$$
$$ \phi(\delta) = \begin{...
- 323
4
votes
1
answer
169
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Universality for reversible classical computation
Is there any way to check whether a set of gates (for example, take the set comprising of the CNOT gate and the Hadamard gate) is universal for reversible classical computation?
I can think of trial ...
- 1,515
7
votes
2
answers
543
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Complexity of $n$-Toffoli with phase difference
I'm interested in the $n$-Toffoli gates with phase differences. I found a quadratic technique in section 7.2 of this paper.
Here's the front page of the paper.
Here's an image of the section that I'm ...
- 101
7
votes
2
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7k
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How to visualize Hadamard gate as $X$-$Z$-$X$ decomposition?
In the book Quantum Computation and Quantum Information by Nielsen and Chuang, chapter 4, exercise 4.4 (pg. 175), the author has asked to express Hadamard gate as product of $R_x$, $R_z$ rotations and ...
- 101
5
votes
2
answers
408
views
How to implement the power of a product of quantum gates as a circuit?
Suppose I have quantum gates (i.e. unitary matrices) $A$ and $B$, and I want to implement $(AB)^x$ in a circuit. If $x$ is integer, I can simply apply $A B$ repeatedly $x$-times. But what if $x$ is a ...
- 2,457
4
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0
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130
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Reducing an ansatz to a shallower circuit
Given a very general hardware efficient ansatz as in Figure:
and say that you already know all the rotation parameter for the gates in the red box, is there any way to build a gate sequence that ...
- 493
3
votes
1
answer
471
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How to create CNOT from an entangling gate and arbitrary single-qubit gates?
I am working on the classical simulation of quantum circuits. I know how to efficiently implement the following entangling gate, which -- in the following paper: https://arxiv.org/pdf/1803.02118 -- ...
4
votes
2
answers
901
views
What gate should one use to perform $R_y$ using a single $R_z$ + Clifford gates?
I know how to perform Rz rotations with the least amount of T gates, eg by using Efficient Clifford+T approximation of single-qubit operators by Peter Selinger. Similarly, one could use H Rz H to ...
- 583
8
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3
answers
2k
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Is it possible to make a Toffoli gate using only CNOTS and ancillas?
I have tried to make a Toffoli gate using only CNOTs and some ancilla qubits but I do not get the unitary. It seems it is not possible without additional gates? What could I do to prove it?
I have ...
- 2,396
4
votes
1
answer
518
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From mathematical notation to quantum circuit, in general
I am learning the basics of quantum computing using Qiskit and I encountered a problem when I tried to solve some of our course exercises. I feel like I am missing an invisible step, the step from ...
- 43
2
votes
3
answers
207
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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
$$U:|...
- 279
3
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1
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677
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How does a general rotation $R_\hat{n}(\theta)$ related to $U_3$ gate?
From eqn. $(4.8)$ in Nielsen and Chuang, a general rotation by $\theta$ about the $\hat n$ axis is given by
$$
R_\hat{n}(\theta)\equiv \exp(-i\theta\hat n\cdot\vec\sigma/2) = \cos(\theta/2)I-i\sin(\...
- 2,408
2
votes
1
answer
159
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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 ...
- 61
2
votes
1
answer
135
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IBM Qiskit QAOA gate implementation question
In section $5.2$ of the QAOA chapter in Qiskit textbook, section $5.2$,
state preparation uses the gate $U_{k,l}(\gamma) = e^{\frac{i \gamma}{2} (1-Z_k Z_l)}$. Later, in section $5.3$, this gate is ...
- 105
7
votes
1
answer
348
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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
$$S=\left(
\begin{array}{cc}
\frac{\sqrt{3}}{2}...
- 135
6
votes
1
answer
2k
views
How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?
How can I construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?
I am stuck in this and any help would be appreciated.
user14639
5
votes
1
answer
238
views
Single-qubit rotations on a subspace within two-qubit unitary
I would like to implement the operation
$$
U(a,b) = \exp\left(i \frac{a}{2} (XX + YY) + i \frac{b}{2} (XY - YX) \right)
$$
($a,b \in \mathbb{R}$) without using Baker-Campbell-Hausdorf expansion, ...
- 7,646
7
votes
3
answers
351
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 ...
- 7,646