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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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 &...
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1answer
61 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
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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
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4answers
475 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
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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
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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
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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
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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
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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
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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
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2answers
324 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
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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
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1answer
243 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
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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
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1answer
82 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
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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
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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
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3answers
734 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
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1answer
58 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
168 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
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
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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
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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
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2answers
164 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
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1answer
62 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
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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
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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}...
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0answers
19 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
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1answer
181 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
57 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
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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
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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
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1answer
183 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
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3answers
113 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 ...
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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
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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
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1answer
290 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
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1answer
466 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
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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
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1answer
141 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
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1answer
176 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
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1answer
94 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
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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
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1answer
260 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
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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
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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
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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
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2answers
198 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
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2answers
155 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 ...
4
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1answer
158 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 ...