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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How to obtain the inner product of the local registers for a superposition state?

For a superposition state $$|{{\Phi }^{+}}\rangle =\frac{1}{\sqrt{4}}\left( |0\rangle |{{U}_{1}}w_1,w_2\rangle +|1\rangle |{{U}_{2}}w_1,w_2\rangle +|2\rangle |w_1,{{U}_{1}}w_2\rangle +|3\rangle |w_1,{{...
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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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1 answer
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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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1 vote
1 answer
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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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