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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23
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4answers
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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 ...
20
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1answer
622 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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6answers
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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}{...
15
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1answer
720 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 ...
14
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2answers
2k 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 ...
14
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1answer
243 views

How does approximating gates via universal gates scale with the length of the computation?

I understand that there is a constructive proof that arbitrary gates can be approximated by a finite universal gate set, which is the Solovay–Kitaev Theorem. However, the approximation introduces an ...
13
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2answers
652 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 ...
13
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2answers
947 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 ...
12
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6answers
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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. ....
12
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3answers
1k 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 ...
12
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2answers
495 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 ...
11
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2answers
5k 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 ...
11
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1answer
179 views

Sampling random circuits vs Solovay-Kitaev compiler

Suppose I want to obtain a gate sequence representing a particular 1 qubit unitary matrix. The gate set is represented by a discrete universal set, e.g. Clifford+T gates or $\{T,H\}$ gates. A well ...
11
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1answer
2k 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?
10
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2answers
573 views

Shortest sequence of universal quantum gates that correspond to a given unitary

Question: Given a unitary matrix acting on $n$ qubits, can we find the shortest sequence of Clifford + T gates that correspond to that unitary? For background on the question, two important ...
10
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1answer
2k 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 ...
10
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1answer
1k 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}...
10
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2answers
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 ...
9
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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 ...
9
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3answers
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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?
9
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1answer
353 views

Basic approximation in Solovay-Kitaev algorithm

I read the Solovay-Kitaev algorithm for approximation of arbitrary single-qubit unitaries. However, while implementing the algorithm, I got stuck with the basic approximation of depth 0 of the ...
8
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3answers
911 views

Is there any method of adding two operators in a circuit?

I am trying to reconstruct the time evolution of a Hamiltonian on the quantum computing simulator, quirk. Ideally I would like to generalise this to any simulator. The unitary matrix is $$U(t)=e^{-...
8
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2answers
448 views

How is it possible to implement unitary operator when its size is exponential in inputs?

A quantum circuit can use any unitary operator. Its matrix is exponential in the number of input bits. In practice how can this ever be possible (aside from operators which are tensor products), i.e. ...
8
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2answers
362 views

Is it possible to realize CNOT gate in 3 dimension?

CNOT gates have been realized for states living in 2-dimensional spaces (qubits). What about higher-dimensional (qudit) states? Can CNOT gates be defined in such case? In particular, is this possible ...
8
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2answers
254 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 ...
8
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1answer
188 views

Nielsen and Chuang's proof for 'approximating arbitrary unitary gates is generically hard'

The following statement is found on the page 199 of Nielsen and Chuang's book (10th Anniversary Edition) in the proof for the fact that 'approximating arbitrary unitary gates is generically hard': ...
8
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1answer
1k views

Decomposition of arbitrary 2 qubit operator

As you know, universal quantum computing is the ability to construct a circuit from a finite set of operations that can approximate to arbitrary accuracy any unitary operation. There also exist some ...
8
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2answers
340 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-...
7
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2answers
2k 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 ...
7
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2answers
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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 ...
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 ...
7
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2answers
388 views

Why is the decomposition of a qubit-qutrit Hamiltonian in terms of Pauli and Gell-Mann matrices not unique?

If I have the $X$ gate acting on a qubit and the $\lambda_6$ gate acting on a qutrit, where $\lambda_6$ is a Gell-Mann matrix, the system is subjected to the Hamiltonian: $\lambda_6X= \begin{pmatrix}0 ...
7
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2answers
1k views

How to obtain Y rotation with only X and Z rotations gates?

Let's say you have a system with which you can perform arbitrary rotations around the X and Z axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the Y axis? ...
7
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2answers
867 views

Simulation vs Construction of Fredkin gate with Toffoli gates

I'm working my way through the book "Quantum computation and quantum information" by Nielsen and Chuang. (EDIT: the 10th anniversary edition). On chapter 3 (talking about reversibility of the ...
7
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1answer
383 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 ...
7
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1answer
819 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,\...
7
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2answers
405 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,...
7
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2answers
287 views

Is there a general method to implement a 'greater than' quantum circuit?

I am interesting in finding a circuit to implement the operation $f(x) > y$ for an arbitrary value of $y$. Below is the circuit I would like to build: I use the first three qubits to encode $|x⟩$, ...
6
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2answers
463 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 ...
6
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3answers
100 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 ...
6
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2answers
250 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-...
6
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3answers
99 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=...
6
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2answers
119 views

Clarification needed: “Simulation” of $e^{-iHt}$ and its time complexity

On page 3 here it is mentioned that: However, building on prior works [32, 36, 38] recently it has been shown in [39] that to simulate $e^{−iHt}$ for an $s$-sparse Hamiltonian requires only $\...
6
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3answers
811 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 ...
6
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1answer
327 views

How to decompose a controlled unitary $C(U)$ operation where $U$ is a 2-qubit gate?

In the vein of this question, say I have a 2-qubit unitary gate $U$ which can be represented as a finite sequence of (say) single qubit gates, CNOTs, SWAPs, cXs, cYs and cZs. Now I need to implement a ...
6
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1answer
373 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 ...
6
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1answer
174 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}...
6
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1answer
118 views

How to factor Ising YY coupling gate into product of basic gates?

Let us consider Pauli YY coupling gate of the following form $$ YY_\phi= \left(\begin{matrix} \cos(\phi) & 0 & 0 & i \sin(\phi) \\ 0 & \cos(\phi) & -i \sin(\phi) & 0 \\ ...
6
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0answers
50 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 ...
5
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2answers
328 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 ...