The Quantum Formalization Drive

This mission develops Chapter 33 of Lecture Notes on Quantum Algorithms by Andrew Childs. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 6 of Quantum Computing: Lecture Notes by Ronald de Wolf. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 7 of Stabilizer Codes and Quantum Error Correction by Daniel Gottesman. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 29 of Lecture Notes on Quantum Algorithms by Andrew Childs. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 3 of Quantum Computing: Lecture Notes by Ronald de Wolf. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 8 of Lecture Notes on Quantum Algorithms by Andrew Childs. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 26 of Lecture Notes on Quantum Algorithms by Andrew Childs. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 18 of Quantum Computing: Lecture Notes by Ronald de Wolf. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 30 of Lecture Notes on Quantum Algorithms by Andrew Childs. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 22 of Lecture Notes on Quantum Algorithms by Andrew Childs. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 12 of Quantum Computing: Lecture Notes by Ronald de Wolf. The chapter's selected results describe the mathematical guarantees below.

The AdS/CFT correspondence relates a theory of quantum gravity in anti-de Sitter space (the bulk) to a quantum field theory without gravity on its boundary (the CFT). A central question is how a local field operator $\phi(x)$ deep in the bulk is represented by boundary operators. Almheiri, Dong and Harlow (JHEP 04 (2015) 163) pointed out a puzzle.

Projective measurement is usually taught together with state update: an observable is measured, a value is obtained, and the conditional state is assigned. These are two different questions. The first is the prediction of the outgoing ensemble from the incoming ensemble and the measured observable alone, before the outcome is known; the second is the…

The parity function tests whether a shallow quantum circuit can combine information from all input bits. Polynomially many ancillary qubits make the measured-output formulation substantially stronger than a restriction to clean final registers. The pinned manuscript supplies the research context.

A quantum channel models how a physical system, used to carry information, is corrupted by its environment. The classical capacity of a channel is the largest rate, in bits per use, at which classical messages can be sent with vanishing error when the sender may encode into arbitrary — possibly entangled — states of many uses and the receiver may measure…

Computing the ground-state energy of electrons in the Coulomb field of fixed nuclei is the basic problem of quantum chemistry. In quantum complexity theory, a problem is QMA-hard if every problem with polynomial-time quantum verification of a quantum witness (the quantum analogue of NP) reduces to it. Hardness was previously known for lattice models with…

Computing the ground-state energy of a molecule — electrons moving in the Coulomb field of fixed nuclei — is the central task of quantum chemistry. Quantum complexity theory asks whether it is hard even for a quantum computer: a problem is QMA-hard if every problem verifiable by a polynomial-time quantum verifier with a quantum witness reduces to it (QMA is…

Parity, $\mathrm{PARITY}_n(x)=x_1\oplus\cdots\oplus x_n$, is the standard test of whether a shallow circuit can aggregate information from all of its inputs. Classically, constant-depth circuits with unbounded fan-in AND/OR gates ($\mathsf{AC}^0$) cannot compute it: Furst, Saxe and Sipser (Math. Systems Theory 1984) and Ajtai (APAL 1983) proved superpolynomial…

The discrete logarithm problem modulo a prime asks, given a prime $p$, a generator $g$ of the multiplicative group modulo $p$, and a nonzero residue $x$, for the exponent $r$ with $g^r\equiv x \pmod p$. Its presumed classical hardness underlies Diffie–Hellman key exchange, ElGamal encryption and the Digital Signature Algorithm. The best classical algorithm known when…

An atom with $N$ electrons is described, with the nucleus fixed at the origin, by a Schrödinger operator on $L^2(\mathbb R^{3N})$ whose potential does not decay in all directions: moving one electron to infinity while the others stay near the nucleus leaves the remaining Coulomb attraction untouched. For this reason the standard route to the essential spectrum…

Shor's 1997 paper (SIAM J. Comput. 26(5), arXiv:quant-ph/9508027) gives a polynomial-time quantum algorithm for factoring integers. The quantum computer does not factor directly: it finds the multiplicative order of an element modulo $n$. The step from order finding to factoring is classical and randomized, and goes back to Miller's 1976 work on primality…

Shor's factoring and discrete logarithm algorithms (Shor 1997) reduce both problems to sampling from the output of a quantum Fourier transform: a register holding a superposition with a hidden period is transformed, then measured, and the measured value carries information about the period. The whole speed-up rests on one engineering fact: for $q = 2^l$ the …

In quantum mechanics the energy levels of a system are the eigenvalues of a self-adjoint Hamiltonian $A$, and for almost every Hamiltonian of interest (atoms with more than one electron, molecules) they cannot be computed in closed form. What can be computed are expectation values $\langle \psi, A\psi\rangle$ of explicit trial functions $\psi$. The min-max…

A quantum particle on a line or half-line, and the radial part of a particle in a spherically symmetric potential, are described by a Sturm–Liouville expression

In quantum mechanics an observable is a self-adjoint operator $A$ on a complex Hilbert space $\mathfrak H$, and the dynamics is the Schrödinger equation $i\,\tfrac{d}{dt}\psi(t) = H\psi(t)$. In finite dimensions the solution is the matrix exponential $e^{-itH}\psi(0)$, computed by diagonalizing $H$. The operators of physics (the Laplacian, Schrödinger operators with…