Questions tagged [inner-product]
For questions about inner products and inner product spaces, including questions about the dot product. An inner product space is a vector space equipped with an inner product. The dot product (seen in multivariable calculus and linear algebra) is a simple example of an inner product—other inner products may be seen as generalizations of the dot product.
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What is the value of $p(+)$?
I know the formula is $p = \left<\psi\right|M_{m}^{\dagger} M_{m}\left|\psi\right>$, where $\left|\psi\right>= \alpha\left|0\right>+\beta\left|1\right>$ and $M_m=\left|+\right> \left&...
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What is the correct order to multiply vectors?
I saw the equation: $$|A\rangle =\sum_i\alpha_i|i\rangle$$
Which could be also written like that:$$|A\rangle=\sum_i|i\rangle\langle i|A\rangle$$
I know that $\langle i|A\rangle=\alpha_i$, but can I ...
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Is the inner product operation commutative or associative?
I am currently reading Quantum Mechanics The Theoretical Minimum by Leonard Susskind.
In the second lecture he says that for a given state of a spin $|A\rangle = a|u\rangle + b|d\rangle$: The ...
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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 ...
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What is the "additive error" of Swap Test?
I'm learning the Swap Test, a quantum circuit to calculate the inner product of two quantum states $|\langle \phi|\psi\rangle|^2 $:
For the error analysis of this quantum circuit, according to Swap ...
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Quantum Algorithm to Determine if two vectors are orthogonal
I have seen some sources use a quantum algorithm to estimate inner products between two states. The algorithm used from this answer is shown here:
But this algorithm has limitations; if the inner ...
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Fidelity concentration bound for random stabilizer states
Let $|\Phi\rangle$ be a normalized vector in $\mathbb{C}^d$ and let $|\psi\rangle$ be a random stabilizer state. I am trying to compute the quantity
$$\mathsf{Pr}\big[|\langle \Phi|\psi \rangle|^2 \...
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Fidelity (overlap) test over reduced density matrices on quantum circuit
The inner product between two quantum states
$\rho(x_1) = U(x_1)|0\rangle\langle 0| U^\dagger(x_1)$ and
$\rho(x_2) = U(x_2)|0\rangle\langle 0| U^\dagger(x_2)$ can be calculated analytically with $Tr[\...
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SWAPing Schmidt vectors
Can anything be said about the inner product of a bipartite entangled state with itself but with the Schmidt vectors swapped? That is, if the Schmidt decomposition of a state is given by
$$\vert \psi \...
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How to obtain the product of the amplitudes of arbitrary basis vectors in a superposition state without measuring?
Suppose there is a superposition state $|{{\Phi }^{+}}\rangle =\sum\limits_{i=0}^{15}{{{\alpha }_{i}}|i\rangle }$, I want to get ${{\alpha }_{i}}\times {{a}_{j}},i\ne j,i,j\in [0,15]$ without ...
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Is there a way to find the inner product between some ground states in a superposition?
Suppose there is a superposition state $|{{\Phi }^{+}}\rangle =\sum\limits_{i=0}^{15}{U({{\theta }_{i}})|i\rangle }$, I want to get $\langle i|U{{({{\theta }_{i}})}^{\dagger }}U({{\theta }_{m}})|m\...
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Inner product of multiple qubit registers
I have read the following statement in some lecture notes:
The inner product of two n-qubit registers is taken by mirrored qubit
pairs. Example: $\lvert ABC\rangle$ and $\lvert abc \rangle$ leading
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How can orthonormal vectors satisfy $\langle i|j\rangle=\delta_{ij}$?
In the book "Quantum Computation and Quantum Information" ("Mike and Ike") - chapter 2, page 66 - I have encountered the following paragraph:
If the vectors i and j are ...
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Is the norm of a inner product symmetric?
I was reading about the Inversion Test and during the derivation (in Machine Learning with Quantum Computers, Schuld and Petruccione) I find the follwing:
Assume we have $|a\rangle = A|0\rangle$ and $|...
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Calculating the Inner Product using Quantum Phase Estimation
I'm following the method laid out in https://arxiv.org/abs/2011.03429 (Page 23 Equations 13-23) to calculate the inner product of two amplitude embedded vectors using Quantum Phase Estimation. I'm ...
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How to calcuate the inner product
How would I calculate the inner product of $|+\rangle|+\rangle$ and $\alpha|00\rangle+\beta|11\rangle$?
I am very new to quantum computing, but I believe for the second problem it would be the ...
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Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$
I am trying to show that $|\langle \psi|_{A} \otimes \langle \phi|_{B}|\theta\rangle_{AB}|^{2}<1$ given $|\theta\rangle$ is an entangled state, and as such has schmidt rank >1. Decomposing it, ...
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Relation between Jordan-Wigner transformation and Hilbert-Schmidt inner product
Given a fermionic Hamiltonian in a matrix form, we can write it as a sum over Kronecker products of Pauli matrices using the Hilbert-Schmidt inner product. However if the same Hamiltonian is given in ...
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How can I prove inequality from 4.66 to 4.67 in Nielson and Chuang's book?
I am reading chapter 4 of Nielson and Chuang's QCQI book.
I cannot prove the inequality from (4.66) to (4.67) in page 195.
That inequality is the following:
$$ |\langle\psi|U^\dagger M|\Delta\rangle|+|...
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How to calculate inner product of quantum states with other method than swap test? [duplicate]
In connection to this question, I am wondering how to calculate value $\langle \psi|\phi \rangle$ for arbitrary quantum states $|\psi\rangle$ and $|\phi\rangle$.
A swap test is able to return only $|\...
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Does the circuit with qubit-wise CZ gates compute the inner product of two states? If not, is there another circuit that does?
I've been searching for a quantum algorithm to compute the the inner product between two $n$-qubit quantum states, namely $\langle\phi|\psi\rangle$, which is in general a complex number.
One can get $|...
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Calculating Dot Product of Two States
I've been reading Peter Wittek's Quantum Machine Learning. In chapter 10.2 of this book, the author explains how we can calculate the dot product of two states:
To evaluate the dot product of two ...
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Compute the squared overlap between different given qubit states
I was checking this problem from the book. And here is an example, but I think it's wrong. If it is not wrong can you please explain how did they derive it?
As per my workout, it should be one. But It ...
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Find inner product of two states given inner product of an orthogonal state
I have a pure quantum state $|i\rangle$ and another state $|\psi\rangle = \frac{1}{\sqrt{2}}(|i\rangle + |j\rangle)$. A state orthogonal to $|\psi\rangle$ is $|\phi\rangle$. Among these states, I know ...
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Discrepancy in inner product between tensor products
I have noticed one identity in case of tensor product from this post. But I can't understand why it is true.
$\langle v_i| \otimes \langle w_j| \cdot |w_k\rangle \otimes |v_m\rangle = \langle v_i|v_m\...
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