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.

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7
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2answers
149 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 ...
4
votes
2answers
81 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
162 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 ...
3
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1answer
130 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) ...
2
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0answers
31 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
44 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
49 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 ...
2
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1answer
41 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
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3answers
630 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
56 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
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1answer
112 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
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1answer
61 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
121 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
82 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
81 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
50 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
82 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
176 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
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0answers
17 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
137 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
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0answers
46 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
102 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
101 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
115 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
105 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
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1answer
38 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
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0answers
54 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
205 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
261 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
104 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
85 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
123 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
75 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
178 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
39 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
171 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 ...
1
vote
1answer
219 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 $R_z(\lambda)$ ...
3
votes
2answers
146 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
137 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 ...
3
votes
1answer
118 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
150 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
78 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?
2
votes
1answer
102 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
45 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
132 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
349 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
108 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
47 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 ...