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Questions tagged [hadamard-test]

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Removing ancilla qubit from circuit with Hadamard test controlled-Z gate

I want to remove the ancilla qubit from the following quantum circuit: Is this possible? The final state of $\newcommand{\ket}[1]{\vert#1\rangle}\newcommand{\bra}[1]{\langle#1\vert}\ket{\psi_1}$ is $\...
upe's user avatar
  • 269
2 votes
0 answers
32 views

Efficient way to compute the $L^1$ norm of quantum state

We all know that the $L^2$ norm $$ ||\psi||_2 = \sqrt{\sum_i |c_i|^2} $$ of a quantum state $|\psi\rangle = \sum_i c_i |i\rangle$ is always equal to $1$. It is possible to compute the $L^1$ norm $$ ||\...
francler's user avatar
  • 181
6 votes
1 answer
107 views

Generalized version of the Hadamard test for $\text{Re} \langle \phi | U | \psi \rangle$

I am wondering if it is possible to generalize the Hadamard test for computing $\text{Re} \langle \phi | U | \psi \rangle$ (different states for left and right operands).
francler's user avatar
  • 181
1 vote
1 answer
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Was Deutsch contemplating a positive-operator valued measurement to distinguish balancedness from constancy?

This is a follow up to a couple of questions on Deutsch's foundational paper on quantum Turing machines. In it, he determines $f(0)\oplus f(1)$ with a single query by measuring a state prepared as $\...
Mark Spinelli's user avatar
2 votes
0 answers
74 views

Given three quantum states, how to compute the triple product of amplitudes $\sum_i u_i v_i w_i$?

Assume I have three quantum states $|u\rangle$, $|v\rangle$ and $|w\rangle$ which can be obtained with three quantum circuits $U$, $V$ and $W$. We know that we can easily estimate the inner product $\...
francler's user avatar
  • 181
6 votes
2 answers
876 views

What is a Hadamard test?

What is a Hadamard test? I have seen this term at many places in video lectures and on various weblinks. A detailed answer on this would be a great help. This is what Wikipedia says, but I really ...
Manu's user avatar
  • 83
1 vote
0 answers
91 views

How to modify the Hadamard test for a non-unitary operator

Assuming I am doing statevector simulations, I need to compute an inner product of the type $$ X_b = \langle\psi | I_0^{\otimes (n-1)} \otimes X | \psi \rangle, $$ where $\psi$ is a generic input ...
francler's user avatar
  • 181
0 votes
0 answers
39 views

Compute the inner product with an operator given by the tensor product of two tensors

Assume I have a quantum register made of $N$ qubits. Assume I want to compute the inner product $$ \langle \psi|I_{n_y} \otimes A_{n_x}| \psi \rangle . $$ Note that I am using statevetor ...
francler's user avatar
  • 181
-1 votes
1 answer
74 views

Help in understanding the algebra of hadamard test

I am new to quantum computing and was looking through the jupyter notebook of Variational Quantum Linear Solver by Qiskit. I came across the hadamard test and was not understanding how it works. It is ...
Vaishnav's user avatar
5 votes
1 answer
394 views

Fidelity (overlap) test over reduced density matrices on quantum circuit

The inner product between two quantum states $\rho(x_1) = U(x_1)|0\rangle\langle 0| U^\dagger(x_1)$ and $\rho(x_2) = U(x_2)|0\rangle\langle 0| U^\dagger(x_2)$ can be calculated analytically with $Tr[\...
incud's user avatar
  • 721
0 votes
2 answers
501 views

How to transform an Hamiltonian operator to a controlled gate (Hadamard test) in Pennylane?

I would like to perform an Hadamard test on a given Hamiltonian operator $\hat{H}$ acting on a 2-qubits system. For instance, suppose $\hat{H}$ can be decomposed into a ...
Constantin Economides's user avatar
10 votes
1 answer
403 views

What is the original reference for the Hadamard test?

The Hadamard test is a widely used routine in quantum computing to compute the real and imaginary part of expectation values of unitary operators. However, all papers I have come across in the ...
bm442's user avatar
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