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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5 answers
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How to derive the CNOT matrix for a 3-qubit system where the control & target qubits are not adjacent?

In a three-qubit system, it's easy to derive the CNOT operator when the control & target qubits are adjacent in significance - you just tensor the 2-bit CNOT operator with the identity matrix in ...
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14 votes
2 answers
3k views

Given a decomposition for a unitary $U$, how do you decompose the corresponding controlled unitary gate $C(U)$?

Suppose we have a circuit decomposition of a unitary $U$ using some universal gate set (for example CNOT-gates and single qubit unitaries). Is there a direct way to write down the circuit of the ...
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13 votes
2 answers
631 views

Quantum XNOR Gate Construction

Tried asking here first, since a similar question had been asked on that site. Seems more relevant for this site however. It is my current understanding that a quantum XOR gate is the CNOT gate. Is ...
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7 votes
2 answers
443 views

Arbitrary powers of NOT and SWAP

The square-root of not and square-root of swap gates are often singled out for discussion of gates displaying important properties relating to quantum computers. How do I define arbitrary (non-...
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4 votes
1 answer
535 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 ...
9 votes
2 answers
3k views

Expressing "Square root of Swap" gate in terms of CNOT

How could a $\sqrt{SWAP}$ circuit be expressed in terms of CNOT gates & single qubit rotations? CNOT & $\sqrt{SWAP}$ Gates Any quantum circuit can be simulated to an arbitrary degree of ...
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7 votes
1 answer
492 views

Quantum XOR Linked List Construction

After getting help here with XNOR & RCA gates I decided to dive into XOR Swaps & XOR linked lists. I was able to find this explanation for quantum XOR Swapping which seems sufficient for the ...
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6 votes
1 answer
513 views

Quantum Ripple Carry Adder Construction

There is an excellent answer to How do I add 1+1 using a quantum computer? that shows constructions of the half and full adders. In the answer, there is a source for the QRCA. I have also looked at ...
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28 votes
6 answers
10k views

How do I build a gate from a matrix on Qiskit?

I'm creating a gate for a project and need to test if it has the same results as the original circuit in a simulator, how do I build this gate on Qiskit? It's a 3 qubit gate, 8x8 matrix: $$ \frac{1}{...
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16 votes
2 answers
9k views

How do you implement the Toffoli gate using only single-qubit and CNOT gates?

I've been reading through "Quantum Computing: A Gentle Introduction", and I've been struggling with this particular problem. How would you create the circuit diagram, and what kind of reasoning would ...
9 votes
3 answers
3k views

How to implement the "Square root of Swap gate" on the IBM Q (composer)?

I would like to simulate a quantum algorithm where one of the steps is "Square root of Swap gate" between 2 qubits. How can I implement this step using the IBM composer?
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12 votes
1 answer
2k views

How to implement a matrix exponential in a quantum circuit?

Maybe it is a naive question, but I cannot figure out how to actually exponentiate a matrix in a quantum circuit. Assuming to have a generic square matrix A, if I want to obtain its exponential, $e^{A}...
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6 votes
1 answer
634 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) ...
18 votes
2 answers
1k views

What is the mathematical justification for the "universality" of the universal set of quantum gates (CNOT, H, Z, X and π/8)?

In this answer I mentioned that the CNOT, H, X, Z and $\pi/8$ gates form a universal set of gates, which given in sufficient number of gates can get arbitrarily close to replicating any unitary ...
16 votes
1 answer
980 views

Obtaining gate $e^{-i\Delta t Z}$ from elementary gates

I am currently reading "Quantum Computation and Quantum Information" by Nielsen and Chuang. In the section about Quantum Simulation, they give an illustrative example (section 4.7.3), which I don't ...
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15 votes
6 answers
10k views

How to construct a multi-qubit controlled-Z from elementary gates?

For the implementation of a certain quantum algorithm, I need to construct a multi-qubit (in this case, a three-qubit) controlled-Z gate from a set of elementary gates, as shown in the figure below. ....
13 votes
2 answers
1k views

Automatic compilation of quantum circuits

A recent question here asked how to compile the 4-qubit gate CCCZ (controlled-controlled-controlled-Z) into simple 1-qubit and 2-qubit gates, and the only answer given so far requires 63 gates! The ...
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13 votes
3 answers
2k views

Approximating unitary matrices

I currently have 2 unitary matrices that I want to approximate to a good precision with the fewer quantum gates possible. In my case the two matrices are: The square root of NOT gate (up to a global ...
  • 4,552
11 votes
3 answers
2k views

Implementation of the oracle of Grover's algorithm on IBM Q using three qubits

I am trying to get used to IBM Q by implementing three qubits Grover's algorithm but having difficulty to implement the oracle. Could you show how to do that or suggest some good resources to get ...
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7 votes
1 answer
1k views

Decomposition of an arbitrary 1-qubit gate into a specific gateset

Any 1-qubit special gate can be decomposed into a sequence of rotation gates ($R_z$, $R_y$ and $R_z$). This allows us to have the general 1-qubit special gate in matrix form: $$ U\left(\theta,\phi,\...
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6 votes
2 answers
2k views

What components are needed to realize a photonic CNOT gate?

In Realization of a photonic CNOT gate sufficient for quantum computation FIG. 1 there is a "scheme to obtain a photonic realization of a CNOT gate with two independent qubits." What ...
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6 votes
2 answers
2k 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 ...
6 votes
1 answer
198 views

Adding a phase to qubit: why is it necessary for arbitrary single qubit gate

By convention, we often write a single qubit gate 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 $e^{i \alpha}$. ...
6 votes
2 answers
443 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 ...
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3 votes
1 answer
180 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?
24 votes
1 answer
827 views

Explicit Conversion Between Universal Gate Sets

I'm interested in the conversion between different sets of universal gates. For example, it is known that each of the following sets is universal for quantum computation: $\{T,H,\textrm{cNOT}\}$ $\{H,...
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11 votes
1 answer
3k views

How can a controlled-Ry be made from CNOTs and rotations?

I want to be able to applied controlled versions of the $R_y$ gate (rotation around the Y axis) for real devices on the IBM Q Experience. Can this be done? If so, how?
11 votes
1 answer
3k views

Implementing a CCCNOT gate using only Toffoli gates

A CCCNOT gate is a four-bit reversible gate that flips its fourth bit if and only if the first three bits are all in the state $1$. How would I implement a CCCNOT gate using Toffoli gates? Assume ...
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9 votes
4 answers
720 views

Minimum number of CNOTs for Toffoli with non-adjacent controls

I want to decompose a Toffoli gate into CNOTs and arbitrary single-qubit gates. I want to minimize the number of CNOTs. I have a locality constraint: because the Toffoli is occurring in a linear array,...
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8 votes
2 answers
435 views

Number of gates required to approximate arbitrary unitaries

If I understand correctly, there must exist unitary operations that can be approximated to a distance $\epsilon$ only by an exponential number of quantum gates and no less. However, by the Solovay-...
8 votes
2 answers
393 views

Construct Controlled-$G^{\dagger}$ from known Controlled-$G$

Let there be a known a scheme (quantum circuit) of Controlled-G, where unitary gate G has G$^†$ such that G≠G$^†$ and GG$^†$=I (for example S and S$^†$, T and T$^†$, V and V$^†$, but not Pauli and H ...
7 votes
3 answers
2k views

Hadamard gate as a product of $R_x$, $R_z$ and a phase

I am having problems with this task. Since the Hadamard gate rotates a state $180°$ about the $\hat{n} = \frac{\hat{x} + \hat{z}}{\sqrt{2}}$ axis, I imagine the solution can be found the following ...
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7 votes
3 answers
3k views

Composing the CNOT gate as a tensor product of two level matrices

I don't understand, why is the control not gate used so often. As far as I understand it, if you apply two 2 level operations on two qubits then you get a 4 x 4 matrix by the tensor product. So how ...
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6 votes
2 answers
549 views

How do we code the matrix for a controlled operation knowing the control qubit, the target qubit and the $2\times 2$ unitary?

Having n qubits, I want to have the unitary described a controlled operation. Say for example you get as input a unitary, an index for a controlled qubit and another for a target. How would you code ...
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5 votes
1 answer
335 views

Decomposition of any 2-level matrix into single qubit and CNOT gates

I saw an example which takes a 2 level matrix. Which is a $8\times8$ matrix that acts non trivially only on 2 levels of only states $|000\rangle$ and $|111\rangle$. The way they do it is by using a ...
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4 votes
1 answer
215 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 ...
4 votes
1 answer
362 views

Square root of NOT as a time-dependent unitary matrix

I want to express the square root of NOT as a time-dependent unitary matrix such that each $n$ units of time, the square root of NOT is produced. More precisely, I want to find a $U(t_0,t_1)$ such ...
3 votes
2 answers
206 views

On the photonic implementation of Shor's code

The above picture comes from this paper. I can see that the standard Shor's code has been re-designed. I have two main doubts: I can't figure out in figure (b) how the setting inputs the state $\...
3 votes
2 answers
161 views

Minimal quantum OR circuit

The quantum OR circuit between $|a\rangle$ and $|b\rangle$ can be made out of 1 Toffoli and 2 CNOT gates, 1 ancillary qubit. Is there any other implementation? Or is this the minimal in the sense of ...
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3 votes
1 answer
116 views

Optimise implementation of a quantum algorithm when an input is fixed

I need to implement a quantum comparator that, given a quantum register $a$ and a real number $b$ known at generation time (i.e. when the quantum circuit is generated), set a qubit $r$ to the boolean ...
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3 votes
1 answer
340 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 $R_z(\lambda)$ ...
2 votes
2 answers
221 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 ...
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2 votes
1 answer
328 views

If CNOTs and single qubit gates are universal then why do we need to prove that controlled U operations can be composed by them as well?

In the book by Chuang and Nielsen they prove that controlled U operations can be made out of CNOTs and single qubit gates. But then they go on to prove that they are universal by showing that every n ...
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1 vote
2 answers
314 views

How does the stated Pauli decomposition for $\operatorname{CP\cdot A\cdot CP}$ arise?

I'm having a bit of trouble understand @DaftWullie's answer here. I understood that the $4\times 4$ matrix $A$ $$ \frac{1}{4} \left[\begin{matrix} 15 & 9 & 5 & -3 \\ 9 & 15 & 3 &...
1 vote
0 answers
62 views

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....
1 vote
0 answers
104 views

PyZX optimisation steps for Clifford circuits

Given the following ZX-diagram It should represent some random Clifford circuit (LC means Local Clifford). As far as I got, any Clifford circuit can be transformed into a ZX-diagram like the above, i....
0 votes
1 answer
800 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 ...