Questions tagged [complexity-theory]

For questions regarding complexity analysis of quantum algorithms and comparisons with complexities of classical algorithms.

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What is the cost of implementing the Quantum Fourier transform in a classical computer?

What is the cost of implementing the Quantum Fourier transform (QFT) in a classical computer? We know we require at least $\log{n}$ depth quantum circuits to do a QFT in a quantum computer, with $n$ ...
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Strong vs weak simulations and the polynomial hierarchy collapse

(Edited to make the argument and the question more precise) An argument for quantum computational "supremacy" (specifically in Bremner et al. and the Google paper) assumes that there exists a ...
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Are there many practical problems for which Grover's algorithm beats the best heuristic classical algorithm?

It's well known that, given an oracle for a function $f$ from a very large set $S$ (of order $N \gg 1$) to $\{0, 1\}$, Grover's algorithm can find an element of $S$ that maps to 1 with $\sim \sqrt{N}$ ...
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Estimating errors in Hamiltonian Simulation paper

I am looking at the paper: Simulating Hamiltonian dynamics with a truncated Taylor series and I am explicitly interested in Eq (15) and (16). These read $$ ||PA |0\rangle |\psi \rangle - |0\rangle ...
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complexity of classical counting algorithm

Does anyone know the solution of Exercise 6.14 of Nielsen and Cheung: Prove that any classical counting algorithm with a probability at least 3/4 for estimating $M$ correctly to within an ...
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Minimum Multi-Degree Polynomials representing Boolean Functions

In the 10th Anniversary Edition of Nielsen and Chuang Quantum Computation and Quantum Information textbook, Chapter 6.7 talks about Black Box algorithm limits. It is given: $f:\{0,1\}^n \...
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187 views

Quantum State Sanitizing

I was reading these lecture notes from Prof. Aaronson about Waltrous's MA protocol for the group non-membership problem. At the end of the description, there's an approach to distinguish if Merlin ...
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141 views

Separating NP from BQP relative to an oracle

I was looking at this lecture note where the author gives an oracle separation between $\mathsf{BQP}$ and $\mathsf{NP}$. He hints at how "standard diagonalisation techniques can be used to make this ...
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Showing that Matrix Inversion is BQP-complete - HHL Algorithm

I am trying to understand an argument that Matrix Inversion is BQP-complete for certain conditions on the matrix. This is explained here on page 39 (this paper is a primer to the HHL algorithm and ...
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Can we amplify BPP algorithms with a random quantum circuit?

Suppose we are given a (univariate) polynomial $P$ of degree $d$, and we wish to determine if $P$ is identically $0$. A standard way to do this is to use a classical PRG to randomly sample $n$ bits, ...
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Illustrating limitations of quantum computing

Can you illustrate why even with a functioning quantum computing energy minimization in an Ising Model simulation, an NP-hard problem, cannot be solved?
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Best query and memory complexity for iterated function

Assume $f(x)$ is n-bit to n-bit function. Let $F(x)$ be defined as $T$ iterations of $f(x)$, i.e. $F(x) = f^T(x)$. Quantum algorithm relies on $F(x)$; it calls it $R$ times. What is the best query ...
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CTCs and information time travel — what non-trivial insights do they lead to?

Context: In quantum complexity theory and quantum information, there are several papers which study the implications of closed timelike curves (CTCs). In 2008, Aaronson and Watrous published their ...
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References on quantum arithmetic circuit complexity

In classical computing, arithmetic circuit complexity is apparently a big topic. But I couldn't find much about the complexity of quantum arithmetic circuits. Almost all references like arXiv:1805....
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Cost of implementing Boolean function quantumly?

Say, I wanted to implement a unitary $U_f$ to compute a Boolean function $f:B_n \to B_n$. This is done by the unitary $$U_f|x\rangle | y \rangle = |x\rangle|y\oplus f(x)\rangle$$ which one can ...
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Complexity analysis of separability in the multipartite case

It's well known that determining whether a bipartite mixed state is separable or entangled is a $\mathsf{NP}$-hard problem under some accuracy estimates (cf. this TCS SE discussion). Now I'm curious ...
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Is it possible to construct an equivalent quantum circuit from a CORDIC-based digital circuit?

DaftWullie mentions an interesting point here: let's assume that we know an efficient classical computation of $f(x)$. That means we can build a reversible quantum computation that runs in the same ...
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Is there a BQP algorithm for each level of the polynomial hierarchy PH?

Note: Cross-posted on Theoretical Computer Science SE. This question is inspired by thinking about quantum computing power with respect to games, such as chess/checkers/other toy games. Games fit ...
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Will quantum computers be able to solve the game of chess?

Will it be possible to use quantum computing to one day solve the game of chess? If so, any estimate as to how many qubits it would require? The game of checkers has already been solved through back ...
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Is there a list of accessible open problems in quantum computing from a theoretical computer science perspective?

(Classical) theoretical computer science (TCS) has a number of outstanding open problems that can easily be instantiated in a manner that is accessible to a wider general public. For example, ...
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Grover's algorithm and Battleship solution

I have read that quantum computers are not known to be able to solve NP-complete problems in polynomial time. However, if you consider a game of Battleship with grid size $X, Y$ and represent this by ...
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Clarification needed for the N&C proof that BQP ⊆ PSPACE

In QCQI by Chuang and Nielsen (page 201), they prove that $\mathsf{BQP} \subseteq \mathsf{PSPACE}$. I can't understand what they say. They write "Supposing the quantum circuit starts in the state $...
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Is quantum computer equivalent to Turing machine with matrix multiplication oracle?

Since quantum computer with $n$ qubits is described by a $2^{n}\times2^{n}$ unitary matrix is it equivalent to an oracle that can do multiplication of large matrix and return $n$ numbers computed ...
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Is there a polynomial quantum algorithm for graph coloring?

Is there a polynomial time and polynomial space quantum algorithm for finding a 4 colouring of any loopless planar graph?
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Why is quantum Fourier transform required in Shor's algorithm?

I’m currently studying the Shor’s algorithm and am confused about the matter of complexity. From what I have read, the Shor’s algorithm reduces the factorization problem to the order-finding problem ...
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What is the complexity of the quantum phase estimation in Grover's algorithm?

Suppose we are using GA (Grover's algorithm) such that we are given it has 2 or more solutions. The search space is of size $N$. We all know Grover's algorithm has, at worst, a time complexity ...
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Does strong error reduction for PostQMA exist?

$\mathsf{PostQMA}$ can be defined as the following (see Morimae-Nishimura and Usher-Hoban-Browne): A promise problem $\mathcal{L}=(\mathcal{L_{yes},L_{no}})$ is in $\mathsf{PostQMA(c,s)}$ if there ...
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Quantum proof for the group non-membership problem

Group non-membership problem: Input: Group elements $g_1,..., g_k$ and $h$ of $G$. Yes: $h \not\in \langle g_1, ..., g_k\rangle$ No: $h\in \langle g_1, ..., g_k\rangle$ Notation: $\...
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How to construct a quantum circuit (QIP system) for the graph non-isomorphism problem?

I'm having some trouble understanding quantum interactive proof systems (QIP systems) and the related circuit constructions. Interactive proof systems model these type of situations: Interactive ...
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Query lower bound for Majority function using the quantum adversary method

Using the quantum adversary lower bound technique, how can one calculate lower bound for Majority function $f:\{0,1\}^n \to \{0,1\}$ such that $f(x)=0$ if $|x|\leq n/2$ else $f(x)=1$, $|x|$ is the ...
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Why doesn't Deutsch-Jozsa Algorithm show that P ≠ BQP?

To my understanding, Deutsch-Jozsa algorithm solves a specific problem in constant time, using a fixed circuit depth, compared to a classical deterministic algorithm, which would require time ...
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Intuitive Proof: BQP ⊆ PP

Promise Problem : It is a pair $$\mathcal{A}=\{\mathcal{A}_{\text{yes}},\mathcal{A}_{\text{no}}\}$$ where $\mathcal{A}_{\text{yes}}$ and $\mathcal{A}_{\text{no}}$ are disjoint sets of inputs (yes ...
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What is recursive Fourier sampling and how does it prove separations between BQP and NP in the black-box model?

Context: I was going through John Watrous' lecture Quantum Complexity Theory (Part 1) - CSSQI 2012. Around 48 minutes into the lecture, he presents the following: No relationship is known between $...
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Classical complexity for Simon's problem

Simon's problem is that you are given a function $f : \{0,1\}^n \to \{0,1\}^n$ such that $f(x)=f(y)$ if and only if $x \bigoplus y$ is either $0^n$ or some unknown $s$. The problem is to find $s$. If $...
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Quantum algorithms for problems outside NP

What is known about quatum algorithms for problems outside NP (eg NEXP-complete problems), both theoretically like upper & lower speedup bounds and various (im)possibility results, as well as ...
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What's the notion of input size for Quantum Verification?

I've been looking into $\mathsf{QPIP}_\tau$ as a complexity class. The following will be a summary of definition 3.12 in Classical Verification of Quantum Computations by Urmila Mahadev. A language ...
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Quantum Zeno Machines

Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that allows a countably infinite number of algorithmic ...
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Clarification needed: “Simulation” of $e^{-iHt}$ and its time complexity

On page 3 here it is mentioned that: However, building on prior works [32, 36, 38] recently it has been shown in [39] that to simulate $e^{−iHt}$ for an $s$-sparse Hamiltonian requires only $\...
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Why is there no $N$ in the time complexity of the QLSP algorithm by Childs et al.?

The paper Quantum linear systems algorithms: a primer by Dervovic et al has this table on page 3: I'm not sure why there's no $N$ in the time complexity of the algorithm by Childs et al. i.e. $\...
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Can we use quantum machines to reduce space complexity of deterministic turing machines?

Can we convert every algorithm in $\text{P}$ (polynomial time complexity for deterministic machines) into a quantum algorithm with polynomial time and $O(\log n)$ quantum bit?
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Does the GLOA have any advantage over the Solovay-Kitaev algorithm?

The Solvay Kitaev algorithm was discovered long before the Group Leaders Optimization algorithm and it has some nice theoretical properties. As far as I understand, both have exactly the same goals: ...
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Relation between $\mathrm{QMA}$ and $\mathrm{P^{QMA}}$

What is the relation between $\mathrm{QMA}$ and $\mathrm{P^{QMA}}$ and how do we prove it? Are these classes equal?
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Quantum Algorithm for God's Number

God's number is the worst case of God's algorithm which is a notion originating in discussions of ways to solve the Rubik's Cube puzzle, but which can also be applied to other combinatorial puzzles ...
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Hilbert space to accurately represent 3x3 Rubik's Cube

What Hilbert space of dimension greater than 4.3e19 would be most convenient for working with the Rubik's Cube verse one qudit? The cardinality of the Rubik's Cube group is given by: Examples 66 ...
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Good metaphors for n-level quantum systems

It seems that a coin flip game is a decent metaphor for a 2-level system. Until 1 of the 2 players picks heads or tails, even if the coin has already been flipped, the win/loss wave form has not yet ...
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Query regarding BQP belonging to PP

I found the following proof of BQP belonging to PP (the original document is here). There is a part of the proof that I have trouble understanding. First, the structure is given below. We try to ...
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What classical public key cryptography protocols exist for which hacking is QMA complete or QMA hard?

Such a public key cryptosystem would be "quantum safe" in the sense that quantum computers cannot efficiently solve QMA hard problems.
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How many operations can a quantum computer perform per second?

I want to know what time complexity is considered efficient/inefficient for quantum computers. For this, I need to know how many operations a quantum computer can perform per second. Can anyone tell ...
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Quantum Walk: Why the need of adding “tail” nodes to the root?

As stated in the question, I have found in several papers (e.g. 1, 2) that in order to perform a quantum walk on a given tree it is necessary to add some nodes to the root $r$, say $r^{'}$ and $r^{"}$....
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What is the actual power of Quantum Phase Estimation?

I have some perplexity concerning the concept of phase estimation: by definition, given a unitary operator $U$ and an eigenvector $|u\rangle$ with related eigenvalue $\text{exp}(2\pi i \phi)$, the ...