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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Circuit to construct a $n$-qubit state which is a superposition of states with only a single qubit being $\lvert1\rangle$ [duplicate]

So the question came up in a book I am working through. Given a circuit with $n$ qubits, construct a state with only $n$ possible measurement results, each of which has only $1$ of $n$ qubits as $1$, ...
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How to construct a quantum gate producing 1 if r divides x, 0 otherwise?

If you have two registers in the state $\frac{1}{2^{n/2}} \sum_{x = 0}^{2^{n/2} - 1} |x\rangle |0\rangle$, how could you construct a gate that produces a superposition of states $|x\rangle|1\rangle$ ...
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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 &...
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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 ...
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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 $|x⟩$, ...
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What quantum gate is XNOR equivalent to?

The standard way to implement a reversible XOR gate is by means of a controlled-NOT gate or CNOT; this is the "standard quantum XOR operation". Physics.Stackexchange Is there a "standard quantum XNOR ...
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How can I decompose a gate into $\{\mathrm{CNOT}, \mathrm{H}, \mathrm{P}(\theta)\}$?

I am working with the set $\{\mathrm{CNOT}, \mathrm{H}, \mathrm{P}(\theta)\}$ where $\mathrm{H}$ is the Hadamard gate, and $\mathrm{P}(\theta)$ is the phase gate with angle $\theta$. I want to build ...
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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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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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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 ...
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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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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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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. ....
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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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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 ...
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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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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-...
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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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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 ...
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Doing maths with controlled-half NOTs

In Quantum Computation with the simplest maths possible there is a section titled "Doing maths with a controlled-half NOT" which covers a reversible-(N)AND circuit with controlled-half NOTs. What ...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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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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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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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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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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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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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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 ...
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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?
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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 ...
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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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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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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 ...
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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 ...

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