Questions tagged [complexity-theory]

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

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Worst Case Asymptotic Complexity of Berstein-Vazirani

How would one determine the worst-case asymptotic complexity ($\theta$) of a Bernstein-Vazirani circuit encoding the secret 1111?
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Is the complexity of a quantum circuit constant in the depth of the circuit?

Take a quantum circuit on $n$ qubits, you have some sequence of gates. You can represent these gates as hermitian matrices, and then with some padding, you could take the product of these matrices, by ...
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How do I construct the oracle for a general Bernstein-Vazirani circuit?

Say I have a secret of length $n$, $s = |x_{n-1}x_{n-2}...x_0 \rangle$. If I want to construct an oracle for this problem would I just insert a CNOT gate on every qubit where the secret's value is 1? ...
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2answers
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Estimating output amplitudes of quantum circuits as GapP functions

Let's fix a universal gate set comprising of a Hadamard gate and a Toffoli gate. Consider an $n$ qubit quantum circuit $U_{x}$, made up of gates from that universal set, applied to initial state $|0^{...
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Degree of $N$-bit Majority function is larger or equal to $N/2$

I am looking to prove that the $N$-bit Majority function $f$, which is 1 if its input $x \in \{0, 1\}^N$ has Hamming weight $> N/2$, and 0 if its input has Hamming weight $\leq N/2$ has degree $\...
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Quantum hardness of XQUATH conjecture

Consider the XQUATH conjectures, as defined here (https://arxiv.org/abs/1910.12085, Definition 1). (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time ...
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Are spin-glass problems NP (-complete)?

It is well known that finding ground states for spin-glass systems (Ising, XY...) is NP-hard (at least as hard as the hardest NP-problems) so that they can be efficiently used to solve other NP ...
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93 views

Complexity of Quantum Satisfiability vs Local Hamiltonians

$k$QSAT$c$ is the promise problem where the input, given in an explicit encoding with finite number of bits, is a set $\{p_{1},p_{2},\ldots p_{m}\}$ of $k$-local projectors over a $n$-qbits register, ...
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60 views

The complexity of LH restricted to projectors

Let's denote $kLP_{c}$ the promise problem where the input, given in an explicit encoding with finite number of bits, is a set $\{p_{1},p_{2},\ldots p_{m}\}$ of k-local projectors over a n-qbits ...
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127 views

How is a promise gap related to a spectral gap?

In linear algebra one often concerns oneself with the spectral gap of a given matrix, which may be defined as the difference between the smallest and second-smallest eigenvalue (or, depending on ...
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140 views

The complexity of LH with constant gap

Kitaev's quantum equivalent of the Cook-Levin Theorem, provides a polynomial time classical reduction from a QMA verification circuit to a sum $H$ of local hamiltonians, such that the least eigenvalue ...
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94 views

How would I theorise a quantum query algorithm in O(1)?

I am currently attempting to solve a problem from Nielsen-Chuang, and I can't seem to figure out how I would do this; I'm trying to implement Grover's algorithm to solve the problem of differentiating ...
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Can quantum computer solve NP-complete problems? [duplicate]

As far as I know, quantum computers are able to solve only some of the NP-Problems in polynomial time, using the Grovers algorithm. I read that if one manages to create a reduction of Grovers ...
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Classical algorithm with complexity similar to Shor's discovered: Are there more efficient quantum algorithms than Shor's?

In the article Fast Factoring Integers by SVP Algorithms the author claims that he discovered classical algorithm for factoring integers in polynomial time. The Quantum Report mentioned here that it ...
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51 views

What does “large-scale universal quantum computation” mean?

I was reading this lecture notes by Sevag Gharibian. He mentioned the following statement: Large-scale universal quantum computer can be built. I do understand what quantum computer is, but I don't ...
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58 views

How do I calculate the number of uses of a unitary $U$ in iterative phase estimation?

How would one go along to calculate the number of uses of an unitary $U$ in Iterative Phase Estimation (IPE) to compare it to the number of uses of $U$ in standard Phase Estimation (Qiskit QPE)?
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Why does QAOA achieve quantum supremacy in an algorithmic sense?

In the paper Quantum Supremacy through the Quantum Approximate Optimization Algorithm the authors claim (last sentence of page 15): "If [...] the QAOA outperforms all known classical algorithms ...
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List of problems in the query complexity model with no super-polynomial quantum speedup

Similar to this list over at cstheory, I'm looking for a list of computational problems in the query complexity model for which it is known that no super-polynomial quantum speedups exist. What are ...
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Properties of QAOA

The QAOA algorithm consists of two elements: The outer loop, basically a classical optimization algorithm The quantum circuit, taking $2p$ parameters (where $p$ is the number of layers, where each ...
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What is the computational complexity of quantum annealing?

Quantum annealing can be thought of as a black box solver that can find approximate solutions to hard optimization problems. For example, D-Wave quantum annealers can approximately solve quadratic ...
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162 views

CS conjecture that Quantum Computer cannot solve NP-complete problems, but Boson Samplers do a #P-hard problem. How is it?

There is something very strange and absurd for me about the power of a quantum computer. Let me briefly states the following facts: Fact 1: theoretical computer scientists believe (very likely to ...
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What is the computational complexity of approximate quantum adders, in terms of big O notation?

I have recently found papers on approximate quantum adders. However, the papers do not seem to mention the computational complexities of their algorithms. What are their complexities, in terms of big ...
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Are almost-Clifford circuits almost easy to simulate?

Circuits consisting entirely of Clifford operations in $\{X, Y, Z, H, S, \text{CNOT} \}$ are "easy" to simulate classically since there is a method that can fully compute such circuits over $...
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Do there exist problems known to be computationally intractable for quantum computer, but tractable for classical computer?

Or alternatively phrased, is it believed that the complexity class P is a complete subset of BQP? Consider the following diagram à la MIT OpenCourseWare, which seems to explicitly state as much.
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What is stopping FACTORING from being BQP-complete?

Classical complexity theory makes much of the study of so-called intermediate problems - that is, problems that are in $\mathsf{NP}$ but are nonetheless not known to be in $\mathsf{P}$ and further not ...
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What kind of boolean functions are faster to compute on qc?

Deutsch-Jozsa algorithm can compute if some function $f : \{0,1\}^n \rightarrow \{0,1\} $ is constant. This goes exponentially faster than on classical computers. If we consider the set of all boolean ...
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Complexity of controlled operations in a two-level unitary operation

In Neilsen and Chuang, chapter 4.5.2 (~p.193), why did the authors come to the conclusion that complexity of operations $C^n(X)$ and $C^n(\tilde{U})$ is $O(n)$? Did they assume using work qubits? If ...
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Hybrid lower bound proof Kaye Laflamme Mosca (lemma 9.3.6)

I am confused about one point in the proof on the lower bounds in Kaye, Laflamme Mosca's lemma 9.3.6. Context: $|\psi_T\rangle$ is the final state of the search algorithm that started on the all-zero ...
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How close is the history state to the ground state in the Kitaev clock construction?

Consider a standard circuit to Hamiltonian reduction in QMA. For example, refer here (Vazirani's lecture notes) or page 235 of here (survey by Gharibian et al). The history state of the Kitaev clock ...
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PreciseQMA = PreciseBQP gives PP = PSPACE

$\text{PreciseBQP}$ is defined as $\text{BQP}$ with inverse exponentially close completeness and soundness bounds (for a better definition, see Section 3.1 here, in the paper by Gharibian et al). ...
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Could random quantum circuits be efficiently approximately simulated?

Google's landmark result last year was to compute a task with a quantum computer that a classical computer could not compute, and they chose random circuit sampling. Part of their justification was ...
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Bound on quantum speedups under various models of complexity

What are the bounds on quantum speedups under the various models of complexity? How big or small can they be? Under the query model, my understanding is that the lower bound is $\Omega(\sqrt{N})$ as ...
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BQP and PH separation

I was reading the Quanta article here which shows that there exists a problem which achieves "oracle separation between BQP and PH". In simple terms, there exists a problem which a quantum ...
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How precise are BQPSPACE measurements?

This is in a similar spirit to another question I asked here. Let's say I am given a $k$-local Hamiltonian $H$. We know that $||H|| \leq 1$. Let the ground state be $|\psi_{0}\rangle$, with energy $E_{...
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28 views

How precise are BQP measurements?

Let's say I am given a Hamiltonian $H$, whose ground state is efficiently preparable. We know that $||H|| \leq 1$. Let that ground state be $|\psi_{0}\rangle$, with energy $E_{0}$. We also know that ...
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102 views

Dirichlet distribution: posteriors and priors of distribution

Let $|\psi\rangle \in \mathbb{C}^{2n}$ be a random quantum state such that $ |\langle z| \psi \rangle|^{2} $ is distributed according to a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution, for $z \in \...
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What is the relationship between quantum circuit and quantum query complexities?

I am trying to ascertain a precise understanding of the relationship between the quantum query model of complexity and the quantum circuit model of complexity. Specifically, is there an established ...
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303 views

Proof that any unitary can be written as $U=e^{-iH}$ with $H$ Hamiltonian with bounded norm

I am looking for a proof that any unitary matrix can be written as: $$U = e^{-iH}$$ where $H$ is some Hamiltonian with bounded norm. That is $$||H||_{2} = O(1).$$
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Proof using hybrid method that inverting a permutation requires exponential queries for BQP machines

Let's say I am given a permutation $\sigma$ that maps $n$ bit strings to $n$ bit strings. I want to output $1$ if $\sigma^{-1}(000\cdots1)$ is even and $0$ if $\sigma^{-1}(000\cdots1)$ is odd. It can ...
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Question regarding soundness bound in QMA versus QCMA separation

I am trying to understand the soundness bound reached in Theorem 4 of this paper, which deals with separating $QMA$ and $QCMA$ with respect to an in-place oracle. To state just the part I am confused ...
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What is the query complexity of the QUBO algorithm?

What is the complexity of the quantum unconstrained binary optimization (QUBO) algorithm in the number of queries to the quantum processor? To clarify, I'm asking about the complexity on quantum ...
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What is known about the quantum version of Schoening's algorithm for 3SAT?

Schoening's algorithm for 3SAT can be converted to a quantum algorithm.  The classical circuit representing a 3SAT expression in CNF form can be converted to a quantum version involving reversible ...
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Is HHL still BQP-complete when the matrix entries are only in {0,1}?

I'm studying BQP-completeness proofs of a number of interesting problems of Janzing and Wocjan, and Wocjan and Zhang. Janzing and Wocjan show that estimating entries of matrix powers $(A^m)_{ij}$ with ...
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Consequences of $MIP^\ast=RE$ Regarding Quantum Algorithms

The (pending-peer review) proof of $MIP^\ast=RE$ in this pre-print has been hailed as a significant breakthrough. The significance of this result is addressed by Henry Yuen (one of the authors) in ...
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Does finding an algorithm that solves an NP-Problem in Polynomial time in a Quantum Computer imply P = NP?

I was wondering if the complexity of a quantum circuit that solves a problem that is in NP implies P=NP?
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Hardwiring the output in black box separation

In this paper, while using a diagonalization argument in Section $5$, the authors write: Fix some enumeration over all $poly(n)$-size quantum verifiers $M_{1}, M_{2},...$ which we can do because the ...
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How do I check if a gate represented by Unitary $U$ is a Clifford Gate?

The Gottesman–Knill theorem states that stabilizer circuits, circuits that only consist of gates from Clifford group, can be perfectly simulated in polynomial time on a probabilistic classical ...
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Can quantum computers be used to solve P = NP

The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be solved quickly. It is one of the seven ...
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Equivalence checking of quantum circuits up to error

Suppose you are given two circuit descriptions $A$ and $B$ where by a circuit description I mean a sequence of gates (in the order they are applied) and the qubits they are applied on. (For the sake ...
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Rings or algebras with many nilpotent elements and efficient computation

Searching the web for '"quantum computer" nilpotent' returns many results, so maybe the question is ontopic for this site. Can a quantum computer solve the following mathematical problem: ...