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.
142
questions
2
votes
1answer
31 views
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
$...
6
votes
1answer
259 views
Is the Solovay-Kitaev theorem relevant for modern hardware?
The Solovay-Kitaev theorem (and more recent improvements) explains how to efficiently compile any 2-qubit unitary into any universal (dense) finite set of gates. My question is if this theorem is ...
1
vote
2answers
31 views
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
$$
XY(\beta, \theta) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 &...
-2
votes
1answer
62 views
Decomposition of U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate [closed]
Can we express U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate
$$ U_1(\lambda) = \begin{pmatrix}1 & 0 \\ 0 & e^{i\lambda}\end{pmatrix}$$
$$ \phi(\delta) = \begin{...
2
votes
2answers
39 views
Optimal decompositions of some standard multi-qubit gates
To have a concrete example in mind: 3-qubit Toffoli gate can be decomposed into 6 $CNOT$s as shown here
I believe this is the most economic decomposition in terms of the number of $CNOT$s used. My ...
2
votes
4answers
482 views
How is a Toffoli gate built without using T gates?
Can someone tell me how to make a Toffoli gate without using T gates? Can we use $R_x$ and $R_y$. If yes, then how?
I tried many circuits but I was unable to create the CCNOT gate out of $R_x$, $R_y$ ...
2
votes
1answer
53 views
Is it possible to decompose $\land(UXU^\dagger)$ in one-qubit operations and only a single $\land(X)$?
Let $U,V$ being any unitary.
Is it possible to decompose $\land(UXU^\dagger)$ in one-qubit operations and only a single $\land(X)$?
Something like the following: $\land(UXU^\dagger) \equiv (\mathbb{I}\...
1
vote
1answer
41 views
Generalized push for $\land_{ab}(X)$ gate
EDIT: In the following I am using the Feynman notation for controlled operations - e.g. $\land_{ab}(X)$ is equivalent to a $CNOT$ with control qubit $q_a$ and target $q_b$. Ultimately, for any single-...
4
votes
0answers
67 views
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 ...
2
votes
2answers
72 views
Is it possible to push back an $H$ gate to a $CZ$ gate?
Given the above scenario. Is it possible to "push back" the $H$ gate operation to occur before $CZ$?
Formally I am looking for some operation $CZ\cdot(U_1\otimes U_2) = H\cdot CZ$.
3
votes
0answers
71 views
How exactly does the QuantumCircuit.decompose() method work?
From what I can understand from the source code, the circuit is converted into a DAG before the decomposition transpiler is performed onto the DAG circuit.
How does converting to a DAG circuit help us ...
7
votes
2answers
233 views
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 ...
6
votes
2answers
332 views
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 ...
5
votes
2answers
177 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 ...
4
votes
1answer
252 views
Transpilation into custom gate set in qiskit
In qiskit, I can transpile a given circuit into a some predefined gate set as follows (just an example)
...
3
votes
0answers
47 views
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 ...
3
votes
1answer
87 views
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
2answers
58 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 ...
3
votes
1answer
66 views
Tool to verify $CNOT$ (or any interacting 2-qubit gate)
Is there any tool to define a circuit and verify if it works as desired?
It would be interesting to find ways of performing interacting gates - e.g. CNOT gate - between non adjacent qubits.
Hence I'd ...
7
votes
3answers
736 views
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
votes
1answer
59 views
Do we need to use an ancillary qubit when decomposing arbitrary $U(2^n)$ gates using Clifford+T universal gate sets?
As I know, we can decompose $U$ without ancilla if it's from special unitary group $SU(2^n)$. Do we need to use ancilla qubit on decomposing arbitrary $n$-qubit $U$ using Clifford+T universal gates ...
3
votes
1answer
169 views
What is Qiskit's Transpiler method for unitary synthesis?
As I could found in here how the transpile works in qiskit, I understood that transpile gets arbitrary Unitary gate $U$ and some set of basis gates as input, and produce some quantum circuit of $U$ ...
3
votes
1answer
78 views
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 ...
2
votes
3answers
139 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
$$U:|...
3
votes
1answer
104 views
How does a general rotation $R_\hat{n}(\theta)$ related to $U_3$ gate?
From eqn. $(4.8)$ in Nielsen and Chuang, a general rotation by $\theta$ about the $\hat n$ axis is given by
$$
R_\hat{n}(\theta)\equiv \exp(-i\theta\hat n\cdot\vec\sigma/2) = \cos(\theta/2)I-i\sin(\...
2
votes
2answers
166 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
64 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 ...
2
votes
1answer
92 views
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 ...
6
votes
1answer
182 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
$$S=\left(
\begin{array}{cc}
\frac{\sqrt{3}}{2}...
0
votes
0answers
20 views
Compiling Quantum Circuits using the Palindrome Transform
This paper shows a way to produce optimal circuits. I haver verified most of them and they are correct except this procedure:
procedure ProduceArray(n)
I cannot ...
3
votes
1answer
183 views
How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?
How can I construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?
I am stuck in this and any help would be appreciated.
2
votes
0answers
58 views
Representation of multiple qubit matrices in Dirac notation
Imagine one wants to represent the and function for any number of qubits in Dirac notation. The and gate flips the target qubit if all the control qubits are in state 1. This is its matrix ...
6
votes
3answers
107 views
How to create the state $\vert 0 \rangle+i \vert 1 \rangle$ using elementary gates?
I am trying to write $|0\rangle+i|1\rangle$ in terms of elementary gates like H, CNOT, Pauli Y, using the IBM QE circuit composer.
I was thinking some kind of combination of H and Y since $Y|0\rangle=...
4
votes
1answer
123 views
Single-qubit rotations on a subspace within two-qubit unitary
I would like to implement the operation
$$
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, ...
3
votes
1answer
184 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
115 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
43 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, ...
6
votes
0answers
64 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
295 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
468 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
113 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
142 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 ...
3
votes
1answer
178 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
96 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
262 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
41 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
237 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 ...
3
votes
1answer
263 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 $R_z(\lambda)$ ...
4
votes
2answers
200 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 ...
