Quantum circuits and error-correcting codes
Fewer T gates on standard circuits, and stabilizer codes with larger distance (placeholder)
Problems
Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
Matching problems
The five-dimensional Watson integral
Find a closed form for the five-dimensional Watson integral W5 ≈ 0.2312616250, which fixes the return probability of a random walk on Z^5.
The mean of the Tracy–Widom distribution
Find a closed form for the mean μ ≈ −1.7710868074 of the Tracy–Widom distribution of the largest GUE eigenvalue.
The critical mass of the Townes soliton
Find a closed form for the critical mass N_c ≈ 11.7008965246 of the Townes soliton, the collapse threshold of the 2D cubic NLS.
Stabilizer rank of tensor powers of the T state
Decide whether the number of stabilizer states needed to write n copies of the T magic state is bounded by a polynomial in n.
Smallest output dimension for a minimum-output-entropy violation
Find the smallest output dimension in which two quantum channels violate additivity of minimum output entropy; it lies between 2 and 182.
The site percolation threshold of the square lattice
Find a closed form for the site percolation threshold of the square lattice, p_c ≈ 0.5927460508; bond percolation has p_c = 1/2.
Quantum capacity of the depolarizing and qubit Pauli channels
Determine the quantum capacity of qubit Pauli channels; even the depolarizing channel's zero-capacity threshold is only known to lie between 0.064956 and 1/12.
Seven qubits protecting a qutrit: does a ((7,3,3)) code exist?
Find a three-dimensional subspace of seven qubits that corrects any single-qubit error, or prove that no ((7,3,3)) code exists.
Additivity of minimum output entropy for qubit channels
Prove or refute that minimum output entropy is additive when one of two channels is an arbitrary, possibly non-unital, qubit channel.
The quantum MDS conjecture
Prove that stabilizer quantum MDS codes on qudits of dimension q have length at most q²+1, apart from one even-q family, or find a longer one.
Simulating partially entangled qubits with one bit of communication
Decide whether shared randomness and one bit of communication reproduce all projective measurements on every partially entangled qubit pair.
Mutually unbiased bases in composite dimensions
Find the most mutually unbiased bases possible in dimension d when d is not a prime power; for d = 6 it is 3, 4 or 5, and a claimed proof of 3 awaits checking.
The monomer-dimer constant of the square lattice
Find a closed form for the entropy per site of monomer-dimer configurations on the square lattice, h ≈ 0.6627989727.
The Madelung constant of rock salt
Find a closed form for the Madelung constant of rock salt, M ≈ 1.7475645946, the alternating lattice sum of unit charges on Z^3.
The Ising-class integral C5
Find a closed form for the Ising-class integral C5 = (4/15)∫ t K0(t)^5 dt ≈ 0.6657598002 of Bailey, Borwein and Crandall.
The hard-square entropy constant
Find a closed form for the hard-square entropy constant κ ≈ 1.5030480825, the growth rate of independent sets in the square grid.
Fewest CNOT gates to prepare the Golay code's logical zero
Prepare the 23-qubit Golay code's logical zero with fewer than 44 CNOT gates, or prove 44 optimal; at depth 7 the record is 45.
Feigenbaum's period-doubling constant
Find a closed form for Feigenbaum's period-doubling constant δ ≈ 4.6692016091, known to 10,000 digits but not proven irrational.
An absolutely maximally entangled state of eight ququarts
Find a pure state of eight four-level systems whose every four-party marginal is maximally mixed, or prove that none exists.