Questions tagged [quantum-computation]
Quantum computing is a model of computation that uses quantum bits instead of classical $0/1$ bits. This allows for the superposition of classically allowable states. Relevant topics include quantum algorithms (e.g. Shor's factoring algorithm), quantum information theory, quantum entanglement, and quantum annealing.
52
questions
113
votes
11
answers
17k
views
On mathematical arguments against Quantum computing
Quantum computing is a very active and rapidly expanding field of research. Many companies and research institutes are spending a lot on this futuristic and potentially game-changing technology. Some ...
34
votes
5
answers
4k
views
What are the strongest arguments for a genuine quantum computing advantage?
Despite having become a fairly mature field with enormous sums of money dumped into research and development, there does not as yet exist a formal proof that quantum computation actually provides an ...
- 756
16
votes
3
answers
1k
views
Are there any known quantum algorithms that clearly fall outside a few narrow classes?
I'm trying to refresh myself on quantum algorithms and have been skimming Childs and van Dam's 2008 RMP paper among other things. From my preliminary surfing it looks like the known quantum algorithms ...
- 15.3k
14
votes
3
answers
3k
views
Will quantum computing kill cryptography ? [closed]
I apologize as this question is not really mathematical, and therefore perhaps not
well-suited for this site. Please feel free to close it if you think it is not. My reason
for asking it here is that ...
- 25.7k
14
votes
2
answers
2k
views
What's known about the relationship about EQP and BQP?
EQP is the class of problems solvable deterministically using a quantum computer in polynomial time - that seems to me to be a good analogue to P, whereas BQP is the quantum analogue of BPP.
It ...
- 1,909
14
votes
1
answer
3k
views
Can Shor's Algorithm be modified to run efficiently on a classical computer?
Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
- 2,539
13
votes
4
answers
2k
views
Quantum algorithms for dummies
I want to try my hand at designing quantum algorithms to solve certain problems. I feel like I understand (for example) how Grover's algorithm and Shor's algorithm work, and I'm excited to apply the ...
- 7,573
13
votes
2
answers
1k
views
Is quantum game theory reducible to classical game theory?
Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:
Superposed initial states,
Quantum ...
- 233
13
votes
1
answer
1k
views
Classification of unitary modular tensor categories (UMTCs)
Context/background:
I'm approaching this topic from the perspective of anyonic systems.
In the study of anyons, one works with fusion categories. Of course, for physicality, we demand that
i) The ...
- 318
13
votes
1
answer
11k
views
Constructing the oracle for Grover's algorithm
For a final project in my class, I decided to try to simulate a quantum computer and implement Grover's algorithm. I followed this excellently written blog post by Craig Gidney, and was successful in ...
- 233
12
votes
3
answers
944
views
Representing SU(3) with 3 ropes in 3 dimensions
The short question is: how exactly is SU(3) realized with ropes?
The long question: There is this idea that deformations of a configuration of three infinitely long, flexible ropes that cross each ...
- 221
11
votes
1
answer
326
views
Unique words in dihedral groups
Suppose $x$ is a word over the alphabet $\{0,1\}$.
Let $a$, $b$ be elements of the group Dih$_k$ for some $k$.
Let $\varphi=\varphi_{a,b,k}$ be the map from words over $\{0,1\}$ to elements of the ...
- 24.6k
11
votes
1
answer
4k
views
How much does a quantum oracle to find a needle in a haystack really cost?
Among the basic algorithms of quantum computations Lov Grover's result on quantum search stands out, both in regards to its intrinsic interest, and for its undisputable elegance.
Grover's algorithm ...
- 7,613
11
votes
1
answer
668
views
Do quantum "Sure-Shor separators" have a natural Veronese/Segre classification? (question inspired by Gil Kalai and Aram Harrow)
Aram Harrow asked: "Is there any place this is written up?"
Update Partly in answer to Aram's question, the thermodynamical properties of varietal dynamical systems now are written-up in ...
- 1,369
9
votes
2
answers
2k
views
Quantum PCP Theorem
Although I think I know the answers to these, I'd just like to collect them all in one place.
What is the quantum PCP theorem, what implications does its proof have for simulation of Hamiltonians and ...
- 93
9
votes
0
answers
306
views
The geometry of lambda calculus?
I stumbled upon "the geometry of quantum computation" --- to quote the abstract:
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding ...
- 1,191
8
votes
1
answer
1k
views
What is the "Tangle" at the Heart of Quantum Simulation?
The following questions generalize and naturalize the question that was originally asked. Provisional answers largely due to Will Sawin are now included.
As was discussed in the question originally ...
8
votes
1
answer
387
views
Are there any unitary matrices which satisfy the Yang-Baxter equation which are universal for quantum computation?
Let $H$ be a finite dimensional hilbert space. Let $L:H\otimes H\rightarrow H\otimes H$ be a unitary transformation. Then the equation
$$(L\otimes I)(I\otimes L)(L\otimes I)=(I\otimes L)(L\otimes I)(I\...
- 27.8k
7
votes
6
answers
3k
views
Presentation of the Clifford group by generators and relations?
The Clifford group $\mathcal{C}_n$ is a matrix group on $\mathbb{C}^{2^n}$ generated by tensor products of the following matrices:
$$
P = \begin{pmatrix} 1 & 0 \\\\ 0 & i\end{pmatrix}
\quad
H =...
- 231
7
votes
3
answers
2k
views
Grover's Quantum Search Algorithm
I am confused about an extremely basic point concerning Grover's quantum search algorithm; my confusion suggests to me that maybe I've missed the entire point.
My understanding of the algorithm is ...
- 22.5k
6
votes
1
answer
186
views
Applications of quantum representations of the mapping class group to quantum computers
Quantum representations of the mapping class group of a surface are certain representations constructed from the data of a TQFT and described, for example, in and 1 and 2.
The following sources 3 ...
- 2,490
6
votes
1
answer
338
views
Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$
The equivalence I describe below is well-known, but I'd like a simple standard reference for it.
Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a ...
- 2,140
5
votes
1
answer
817
views
What role do Hecke operators and ideal classes perform in "Quantum Money from Modular Forms?"
Cross-posted on QCSE
An interesting application of the no-cloning theorem of quantum mechanics/quantum computing is embodied in so-called quantum money - qubits in theoretically unforgeable states. ...
- 2,143
5
votes
1
answer
315
views
Set of physical states of FQHE on closed Riemann surface = ?
Disclaimer. One might argue that my question is off topic as it is clearly a question about physics... But I'd like a mathematically phrased answer,
and I expect that only a mathematician can offer an ...
- 42.5k
5
votes
0
answers
289
views
Quantum P vs NP equivalent problem
If $P = NP$, does it follow that $BQP = NP^{BQP}$?
I came up with this question when I was thinking about how $P = NP$ can be described as "does every decision problem where a proof for YES can be ...
- 51
4
votes
1
answer
140
views
An introductory reference for tensor networks
I found a good reference on Tensor Networks: https://arxiv.org/abs/1912.10049. But I need an introductory reference with detailed proofs on Tensor Networks. Do you know another reference?
- 109
4
votes
1
answer
561
views
Are all quantum cellular automata invertible & representable?
A little background: As far as I know there is no standard definition of a quantum cellular automaton in the literature. Different authors use different definitions. Here I propose my own definition (...
- 1,368
3
votes
1
answer
305
views
Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?
Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows:
$$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
- 489
3
votes
2
answers
1k
views
Amplitude amplification as a quantum walk algorithm
This is a followup to an earlier question on a taxonomy for quantum algorithms in which I ultimately concluded in a comment that all known nontrivial quantum algorithm speedups (in Jordan's quantum ...
- 15.3k
3
votes
1
answer
1k
views
Bounding the von Neumann entropy of a density matrix with the Hilbert-Schmidt norm
Question
Suppose I have a $D$-dimensional density matrix $\rho_0$
$\rho_0^\dagger = \rho_0 \quad, \quad \mathrm{Tr} \rho_0 = 1 \quad, \quad \rho_0 > 0,$
with a known spectrum $\{\lambda_i^0\}$ and ...
- 846
3
votes
1
answer
94
views
Question about equivalence of two expressions for the Quantum Fourier transformation
The Quantum Fourier transformation on $n$ qubits is just the discrete Fourier transformation,
$$
|j \rangle \mapsto \frac 1 {\sqrt 2^n}\sum_{k=0}^{2^n-1}e^{2\pi ijk/2^n}|k\rangle.
$$
In binary ...
- 4,304
3
votes
1
answer
160
views
Approximations of the spectral radii of completely positive superoperators
Let $V$ be a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear operators from $V$ to $V$. An operator $\mathcal{E}:L(V)\rightarrow L(V)$ is said to be positive ...
- 27.8k
3
votes
0
answers
107
views
Could a quantum computer factor $N=p\times q$ using Hadamard transforms on $x^2\bmod N$ (instead of Fourier transforms on $a^x\bmod N$)?
In Classically verifiable quantum advantage from a computational Bell test, Kahanamoku-Meyer, Choi, Vazirani, and Yao propose using $x^2 \bmod N$ in an interactive proof-of-quantumness. This is a two-...
- 2,143
3
votes
0
answers
130
views
determine degree of boolean polynomial given as black box
I searched a lot but couldn't find a good resource that addresses this question. Given a boolean polynomial with $n$ boolean variables as a black box, what is the most efficient way to compute its ...
- 103
3
votes
0
answers
191
views
Normalizer of SU(2) x SU(2) in SU(4) [closed]
What is the normalizer of SU(2) x SU(2) in SU(4) or how would I find it?
Reason for the question: with 2 qubits, if I was interested in conjugation of 2-qubit gates with generic SU(2) elements, ...
- 31
3
votes
0
answers
75
views
Functional characterization of local correlation matrices?
Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...
- 1,816
3
votes
0
answers
169
views
Efficient implementation of the Clifford group for $n$ qubits
I'm looking for an efficient implementation of the Clifford group $\mathcal{C}_n$
of $n$ qubits.
The Clifford group $\mathcal{C}_n$ has stucture $(2_+^{1+2n} \circ C_8).Sp(2,n)$,
where $2_+^{1+2n}$ ...
- 558
3
votes
0
answers
72
views
Multilinear maps that preserve unitarity
Let $M_1, M_2, M_3$ be spaces of square complex matrices, respectively acting on finite-dimensional Hilbert spaces $V_1, V_2$, and $V_3 = V_1 \otimes V_2$. Consider bilinear maps
$$\phi: M_1 \times ...
- 31
3
votes
0
answers
754
views
Why isn't Montgomery modular exponentiation considered for use in quantum factoring?
It is well known that modular exponentiation (the main part of an RSA operation) is computationally expensive, and as far as I understand things the technique of Montgomery modular exponentiation is ...
- 15.3k
2
votes
1
answer
207
views
Is quantum turing machine equivalent classical turing machine? [closed]
I have the question if quantum computation is intrinsecally different to a classic computation. Thank you all!!
2
votes
2
answers
672
views
How can I get all the good items using quantum search algorithm?
One can get a superposition of all good item using quantum search algorithm in $O$($\sqrt{N}$ ) time, but how one can get all the good items using quantum search algorithm?
I found that all the good ...
- 21
2
votes
1
answer
150
views
Explain how to infer a density matrix from the statistics of quantum measurements
This question follows the "Probabilistic Simulation of Quantum Circuits with the Transformer" paper by Carrasquilla et al. In the Formalism section on page 2 the authors state that ...
- 135
1
vote
1
answer
344
views
Fixed point of quantum operations
A quantum operation is defined as
\begin{equation}
\varepsilon(\rho)=\sum_{k}M_k\rho M_k^{\dagger}
\end{equation}
where $\varepsilon(\rho)$ takes an initial state $\rho$ to some final state $\rho'$ ...
- 35
1
vote
1
answer
80
views
Complexity classes generated by differential equations
The quantum computer can be represented as a turing machine that sets up initial conditions for Schrodinger-like equation plus a fast ($O(1)$) solver for that equation.
Is there a general study for ...
- 111
1
vote
1
answer
302
views
How to create a quantum algorithm that produces 2 n-bit sequences with equal number of 1-bits?
I am interested in a quantum algorithm that has the following characteristics:
output = 2n bits OR 2 sets of n bits (e.g. 2 x 3 bits)
the number of 1-bits in the first set of n-bits must be equal to ...
- 121
1
vote
0
answers
29
views
Is every nearly rank 1 doubly stochastic superoperator the product of pairwise averagings of unitary channels?
Suppose that $\mathcal{X}$ is a finite dimensional complex Hilbert space. Let $L(\mathcal{X})$ denote the collection of all linear mappings from $\mathcal{X}$ to $\mathcal{X}$. We say that a linear ...
- 27.8k
0
votes
1
answer
226
views
Is there a quantum Bayes rule?
This question has been bothering me for a while. Wading through the internet hasn't turned up any answers that I have been able to understand.
First some motivation: Let $S = \{s_1,s_2,s_3\}$ be a ...
- 3,710
0
votes
2
answers
617
views
A non-associative three-valued logic
There are three elements: x, y, z and a relation C:
x C y, y C z, z C x, x C x, y C y, z C z.
Let us introduce two binary operations with respect to the C: "the leftmost" (L) ...
- 211
0
votes
1
answer
322
views
QUBO formulation of a discrete-variable Genetic Algorithm optimization problem
I am facing a non-linear, discrete optimization problem, which I can formulate in this abstract manner: I have a certain non-analytic real-valued function $f$ depending on a set of parameters $ \theta\...
0
votes
0
answers
26
views
Estimating ground state energy of $n$-qubit $2$-local Hamiltonian $H$ with known coefficients
Suppose we have an $n$-qubit $2$-local Hamiltonian $H$ with known coefficients. The eigenvalues of $H$ lie in $[0,1)$ and can all be written exactly with $[2 \log_2n]$ bits of precision. You would ...
