All records

212 records

• Linear-size exact quantum Fourier transform

  Can the exact quantum Fourier transform on \(n\) qubits be implemented with \(O(n)\) one- and two-qubit gates?

  • Quantum Algorithm
  • Quantum circuit complexity
  Unsolved
• Unrestricted quantum time–space tradeoff for collision finding

  Does every unrestricted quantum collision-finding algorithm satisfy \(T^2S=\Omega(N\log N)\)?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Subquadratic total quantum gate complexity of RSA factoring

  Can every balanced RSA semiprime be factored with polynomially subquadratic total quantum gate complexity?

  • Quantum Algorithm
  • Computational complexity and computability
  • Quantum circuit complexity
  Unsolved
• High-dimensional crossover of optimized port-based teleportation

  What is the high-dimensional crossover fidelity of optimized deterministic port-based teleportation?

  • Quantum Communication
  • Channel simulation
  • One-shot and finite-blocklength bounds
  Unsolved
• Quantum Mrs. Gerber lower bound for information combining

  Does the quantum Mrs.

  • Quantum Communication
  • Matrix and entropy inequalities
  Unsolved
• Sparse Hamiltonian learning without short-time control

  Can every polynomially sparse Hamiltonian be learned with Heisenberg-limited total evolution time when every oracle call has a fixed minimum duration?

  • Quantum Metrology
  • Quantum Algorithm
  • Quantum estimation
  • Computational complexity and computability
  Unsolved
• Sublinear logical quantum space for RSA factoring

  Can every balanced RSA semiprime be factored in polynomial time using sublinear logical quantum space?

  • Quantum Algorithm
  • Computational complexity and computability
  • Quantum circuit complexity
  Unsolved
• Sampling weakly depolarized constant-depth random circuits

  Can weak final-layer depolarizing noise make constant-depth two-dimensional Haar-random circuits classically samplable to inverse-polynomial total-variation error?

  • Quantum Algorithm
  • Random circuit sampling
  • Computational complexity and computability
  • Quantum supremacy
  Unsolved
• Optimal dimension-free copy complexity of shadow tomography

  Can list-dependent shadow tomography always achieve dimension-free copy complexity \(O(\log M/\varepsilon^2)\)?

  • Quantum Metrology
  • Quantum estimation
  • One-shot and finite-blocklength bounds
  Unsolved
• Quantum validity of the Zhang–Yeung inequality

  Does the four-party Zhang–Yeung inequality hold for the von Neumann entropies of every finite-dimensional quantum state?

  • Quantum Communication
  • Matrix and entropy inequalities
  Unsolved
• Multiplicative sparse-access lower bound for quantum linear systems

  Does sparse-oracle quantum linear-system solving require \(\Omega(\kappa\sqrt{s}\log(1/\varepsilon))\) queries in the worst case?

  • Quantum Algorithm
  • Quantum linear algebra
  • Computational complexity and computability
  Unsolved
• Hilbert–Schmidt inradius of the multiqubit stabilizer polytope

  Is every \(n\)-qubit state of purity at most \(1/(2^n-1/2)\) a convex mixture of stabilizer states?

  • Quantum Resource Theory
  • Quantum magic
  Unsolved
• Distillation of every Wigner-negative depolarized Strange state

  Can every Wigner-negative state in the depolarized qutrit Strange-state family be distilled arbitrarily accurately by catalyst-free stabilizer operations?

  • Quantum Resource Theory
  • Quantum magic
  • Resource conversion
  Unsolved
• Conditional-correlation decay in amplitude-damped random circuits

  Do computational-basis outputs of one-dimensional Haar-random circuits with fixed amplitude damping have conditional mutual information that decays exponentially with separation, uniformly in circuit depth?

  • Quantum Algorithm
  • Random circuit sampling
  • Computational complexity and computability
  Unsolved
• Unknown-structure Hamiltonian learning from Gibbs states at all temperatures

  Can every bounded-degree local Hamiltonian with unknown interaction support be learned efficiently from copies of its Gibbs state at any fixed inverse temperature?

  • Quantum Metrology
  • Quantum Algorithm
  • Quantum estimation
  • Computational complexity and computability
  • Quantum thermodynamics
  Unsolved
• Optimal single-copy tomography of fermionic Gaussian states

  Can arbitrary mixed fermionic Gaussian states be learned to trace distance \(\varepsilon\) from \(\Theta(m^2/\varepsilon^2)\) copies without measurements across copies?

  • Quantum Metrology
  • Quantum estimation
  • Gaussian quantum information
  • One-shot and finite-blocklength bounds
  Unsolved
• Uniform modified log-Sobolev constant for one-dimensional Gibbs samplers

  Do normalized one-dimensional Chen–Kastoryano–Gilyén Gibbs samplers have a positive modified log-Sobolev constant uniformly in system size at every fixed finite temperature?

  • Quantum Resource Theory
  • Quantum Algorithm
  • Quantum thermodynamics
  • Quantum state preparation
  • Quantum relative entropy
  Unsolved
• Qutrit magic-distillation thresholds beyond the ternary Golay code

  Does any finite qutrit stabilizer-code protocol distill depolarized Strange states above the ternary Golay code’s threshold?

  • Quantum Resource Theory
  • Quantum Error Correction
  • Quantum magic
  • Quantum coding theory
  • Resource conversion
  Unsolved
• Pseudorandom unitaries from ordinary random two-qubit circuits

  Does a polynomial-length random walk generated by ordinary two-qubit gates form a pseudorandom-unitary ensemble?

  • Quantum Cryptography
  • Quantum Algorithm
  • Quantum circuit complexity
  • Computational complexity and computability
  Unsolved
• Exact spectral gap of random Pauli rotations

  Is the exact Haar-mixing spectral gap of uniformly random Pauli rotations equal to \(2^n(2^n-3)/(8(4^n-1))\) for every \(n\geq4\)?

  • Quantum Algorithm
  • Random circuit sampling
  • Quantum circuit complexity
  Unsolved
• Optimal asymptotic distillation yield of dephased T states

  What is the optimal catalyst-free asymptotic \(T\)-state yield of a dephased single-qubit \(T\) state?

  • Quantum Resource Theory
  • Quantum magic
  • Resource conversion
  • Additivity and regularization
  Unsolved
• Single-copy identity testing of local Gibbs states

  What is the minimax copy complexity of identity testing two unknown \(k\)-local Gibbs states using only measurements on individual copies?

  • Quantum Metrology
  • Quantum Algorithm
  • Quantum hypothesis testing
  • One-shot and finite-blocklength bounds
  • Quantum thermodynamics
  Unsolved
• Optimal universal eight-copy concentration to a CCZ state

  What is the optimal success probability for a fixed catalyst-free stabilizer protocol that converts eight copies of an arbitrary unknown pure qubit into an exact \(|CCZ\rangle\) state?

  • Quantum Resource Theory
  • Quantum magic
  • Resource conversion
  • One-shot and finite-blocklength bounds
  Unsolved
• CNOT-count and depth frontier for quantum Golay state preparation

  What is the exact CNOT-count and depth frontier for ancilla-free Clifford preparation of the logical zero state of the \([[23,1,7]]_2\) quantum Golay code?

  • Quantum Error Correction
  • Quantum Algorithm
  • Quantum coding theory
  • Quantum state preparation
  • Quantum circuit complexity
  Unsolved
• Quantum erasure-channel upper bound for information combining

  Does the quantum erasure-channel upper information-combining bound hold for two independent uniform bits with arbitrary quantum side information?

  • Quantum Communication
  • Matrix and entropy inequalities
  Unsolved
• Optimal precision dependence of diamond-norm channel tomography

  Does tomography of an arbitrary \(d\)-dimensional quantum channel in diamond norm require and suffice with \(\Theta(d^4/\varepsilon^2)\) channel uses?

  • Quantum Metrology
  • Quantum estimation
  • One-shot and finite-blocklength bounds
  Unsolved
• Endpoint quantum KKL inequality for Boolean observables

  Does the Montanaro–Osborne \(L^2\) quantum KKL inequality hold for every Boolean quantum observable?

  • Quantum Algorithm
  • Computational complexity and computability
  • Matrix and entropy inequalities
  Unsolved
• One-way state generation from private Hamiltonian phase states

  Do private-architecture Hamiltonian phase states yield a one-way state generator for explicit polynomial parameters?

  • Quantum Cryptography
  • Computational complexity and computability
  • Quantum state preparation
  Unsolved
• Exact constant-bond PEPS mixtures for two-dimensional Gibbs states

  Is every fixed-temperature Gibbs state of a bounded finite-range qubit Hamiltonian on the square lattice an exact convex combination of PEPS with system-size-independent bond dimension?

  • Quantum Resource Theory
  • Quantum thermodynamics
  • Quantum state preparation
  Unsolved
• Scalable pseudorandom unitaries with an independent security parameter

  Can pseudorandom-unitary security scale independently of Hilbert-space dimension while construction uses only polynomially many oracle queries?

  • Quantum Cryptography
  • Computational complexity and computability
  • Quantum circuit complexity
  Unsolved
• Aaronson–Ambainis conjecture

  Must every bounded low-degree real polynomial on the Boolean cube with non-negligible variance have a variable of inverse-polynomial influence?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Average-case approximation hardness of random-circuit output probabilities

  Does there exist a fixed family of \(n\)-qubit circuit layouts with \(m=\operatorname{poly}(n)\) one- and two-qubit gates for which the following task is \(\#\mathrm{P}\)-hard?

  • Quantum Algorithm
  • Computational complexity and computability
  • Quantum supremacy
  • Random circuit sampling
  Unsolved
• Quantum query complexity of Triangle Finding

  What is the bounded-error quantum query complexity of finding a triangle in an \(n\)-vertex graph given oracle access to its adjacency matrix?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Asymptotic growth of the stabilizer rank of T-state tensor powers

  Does the exact stabilizer rank of the tensor powers of the single-qubit magic state \(|T\rangle\) grow polynomially in the number of copies, or is it not polynomially bounded?

  • Quantum Algorithm
  • Quantum magic
  • Quantum circuit complexity
  • Computational complexity and computability
  Unsolved
• Squashed entanglement of qubit Bell-diagonal states

  Is there a closed-form formula for the squashed entanglement \(E_{\mathrm{sq}}\) of an arbitrary two-qubit Bell-diagonal state, or a certified approach to determining it numerically?

  • Quantum Resource Theory
  • Entanglement measures
  • Bell-diagonal states
  Unsolved
• Wigner entropy conjecture

  Does every finite-energy single-mode state with a nonnegative Wigner function have Wigner entropy at least \(1+\ln\pi\)?

  • Quantum Communication
  • Matrix and entropy inequalities
  • Gaussian quantum information
  Solved
• Diamond-smoothed max-relative-entropy AEP for quantum channels

  Does the max-relative entropy of finite-dimensional quantum channels satisfy an asymptotic equipartition property under uniform diamond-norm smoothing?

  • Quantum Communication
  • Quantum relative entropy
  • One-shot and finite-blocklength bounds
  Unsolved
• Fixed-error parallel Stein lemma for quantum channels

  Let \(\mathcal N,\mathcal M:\mathcal L(A)\to\mathcal L(B)\) be quantum channels on finite-dimensional systems.

  • Quantum Metrology
  • Quantum Communication
  • Channel discrimination
  • Quantum hypothesis testing
  • Quantum relative entropy
  • Strong converses
  Unsolved
• Eight-ququart AME state and the ((7,4,4)) ququart MDS code

  Does an absolutely maximally entangled state of eight ququarts exist?

  • Quantum Resource Theory
  • Quantum Error Correction
  • Absolutely maximally entangled states
  • Quantum coding theory
  Unsolved
• Asymptotically good quantum locally testable codes

  Does there exist an infinite family of qubit CSS codes with linear rate and linear distance whose check matrices have uniformly bounded row and column weights and constant soundness?

  • Quantum Error Correction
  • Quantum LDPC codes
  • Quantum coding theory
  Unsolved
• LOCC entanglement cost of Werner states

  What is the entanglement cost of the Werner state \(\rho_{d,p}\) for every integer \(d\geq2\) and every antisymmetric weight \(1/2<p\leq1\), and in particular, does it equal the entanglement of formation?

  • Quantum Resource Theory
  • Entanglement cost
  • Local operations and classical communication
  Unsolved
• Optimal dimension of nine-qubit distance-three codes

  Is the known non-additive \(((9,12,3))_2\) code of largest possible dimension, or does a nine-qubit code of dimension thirteen and distance three exist?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Many-copy Bell nonlocality of every entangled state

  Does every entangled bipartite state become Bell nonlocal under collective local measurements on some finite number of copies?

  • Quantum Resource Theory
  • Superactivation
  • Bell nonlocality
  • Quantum separability
  Unsolved
• Good quantum LDPC codes with a non-Clifford transversal CCZ gate

  Do there exist triples of qubit CSS LDPC codes with linear combined rate and linear distance on which the transversal CCZ gate induces a non-Clifford logical operation?

  • Quantum Error Correction
  • Quantum LDPC codes
  • Quantum coding theory
  Unsolved
• Superactivation of bipartite classical secret-key rates

  Can two classical sources, each with zero secret-key rate between two honest parties, yield a positive secret-key rate when used jointly?

  • Quantum Cryptography
  • Superactivation
  • Secret-key distillation
  Unsolved
• LOCC entanglement cost of isotropic states

  What is the entanglement cost of the isotropic state \(\sigma_{d,F}\) for every integer \(d\geq2\) and every fidelity \(1/d<F<1\)?

  • Quantum Resource Theory
  • Entanglement cost
  • Local operations and classical communication
  Unsolved
• Minimal block length of permutation-invariant qubit codes

  Is the least number of physical qubits on which a two-dimensional permutation-invariant code corrects arbitrary errors on \(t\) qubits equal to \(3t^2+3t+1\) for every integer \(t\geq1\)?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Superactivation of distillable secret key

  Can two bipartite quantum states with zero distillable secret key yield positive distillable secret key when used jointly?

  • Quantum Cryptography
  • Quantum Resource Theory
  • Superactivation
  • Secret-key distillation
  Unsolved
• Efficient determination of the Clifford-hierarchy level of bounded-degree permutation gates

  For each fixed degree bound \(d\geq2\), is there a deterministic polynomial-time algorithm that computes the Clifford-hierarchy level of an \(n\)-qubit permutation gate from degree-at-most-\(d\) algebraic normal forms of the permutation and its inverse, or reports…

  • Quantum Algorithm
  • Clifford hierarchy
  • Computational complexity and computability
  Unsolved
• Weight-four generators for an [[11,3,3]] stabilizer code

  Does there exist a qubit stabilizer code with parameters \([[11,3,d]]\) and \(d\geq3\) whose stabilizer group is generated by eight independent commuting Pauli operators of weight at most four?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Superactivation of two-way secret-key capacity

  Can two quantum channels with zero two-way-assisted secret-key capacity have positive secret-key capacity when used jointly?

  • Quantum Cryptography
  • Quantum Communication
  • Superactivation
  • Secret-key distillation
  Unsolved
• Smallest qubit number for a non-semi-Clifford third-level gate

  What is the smallest number of qubits on which the third level of the Clifford hierarchy contains a gate that is not semi-Clifford?

  • Quantum Algorithm
  • Clifford hierarchy
  Unsolved
• Optimal dimension of seven-qubit distance-two codes

  For an exact seven-qubit code that detects every single-qubit error, what is the largest possible code-space dimension: is \(K_{\max}^{(2)}(7,2)\) equal to \(24\), \(25\), or \(26\)?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Linear-size logarithmic-depth encoders for good quantum LDPC codes

  Does some family of asymptotically good qubit CSS LDPC codes admit unitary encoding circuits with linearly many gates and logarithmic depth?

  • Quantum Error Correction
  • Quantum Algorithm
  • Quantum LDPC codes
  • Quantum circuit complexity
  Unsolved
• Two-error perfect quantum codes over non-prime-power local dimensions

  Does a nontrivial pure perfect quantum code correcting arbitrary errors on two qudits exist for some local dimension \(q\) with at least two distinct prime factors?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Finite-energy superactivation of Gaussian amplifiers and additive-noise channels

  Is there a single-mode phase-insensitive Gaussian amplifier or additive-noise channel of zero finite-energy quantum capacity that acquires positive finite-energy quantum capacity when used in parallel with a two-mode positive-partial-transpose Gaussian channel…

  • Quantum Communication
  • Superactivation
  • Quantum capacity
  • Bosonic channels
  Unsolved
• Distance of constant-rate quantum XYZ product codes

  Do there exist quantum XYZ product codes, built from classical parity-check matrices with uniformly bounded row and column weights, that have constant rate and minimum distance of order \(N^{2/3}\) on \(N\) physical qubits?

  • Quantum Error Correction
  • Quantum LDPC codes
  • Quantum coding theory
  Unsolved
• Minimal permutation-invariant qubit code for two errors

  Is nineteen the least number of physical qubits on which a two-dimensional code contained in the fully symmetric subspace corrects every error on at most two qubits?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Square-root distance bound for weight-four stabilizer codes

  Is there a universal constant \(C>0\) such that every qubit stabilizer code that encodes at least one logical qubit, and whose stabilizer group is generated by Pauli operators of weight at most four, has distance at most \(C\sqrt n\) on \(n\) physical qubits?

  • Quantum Error Correction
  • Quantum LDPC codes
  • Quantum coding theory
  Unsolved
• Channels superactivated by antidegradable helpers

  What necessary and sufficient structural conditions characterize the finite-dimensional channels of zero quantum capacity, and separately those of zero private capacity, that acquire a positive capacity of the same kind when used in parallel with some antidegradable…

  • Quantum Communication
  • Quantum Cryptography
  • Superactivation
  • Quantum capacity
  • Private capacity
  • Channel degradability
  Unsolved
• Semi-Clifford structure of two-qudit hierarchy gates above the third level

  Is every two-qudit gate in every level \(k\geq4\) of the Clifford hierarchy semi-Clifford when the local dimension is an odd prime?

  • Quantum Algorithm
  • Clifford hierarchy
  Unsolved
• Non-additive ((6,2,3)) qubit code with transversal T

  Does there exist a genuinely non-additive two-dimensional six-qubit code of distance three on which a tensor product of single-qubit unitaries implements the logical \(T\) gate exactly?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Highest Clifford-hierarchy level of controlled gates with higher-level targets

  What is the highest Clifford-hierarchy level that a controlled gate can reach when its target lies in a given level \(k\geq3\) and has a given Pauli periodicity, and is that level bounded independently of the number of target qubits?

  • Quantum Algorithm
  • Clifford hierarchy
  Unsolved
• Zero two-way quantum capacity of endpoint Holevo–Werner channels

  Is the two-way assisted quantum capacity of the Holevo–Werner channel \(\mathcal W_d\) zero for every integer \(d\geq3\)?

  • Quantum Communication
  • Quantum Resource Theory
  • Quantum capacity
  • Bound entanglement
  • Entanglement distillation
  Unsolved
• Geelen’s simulation conjecture for vertex-minor-closed graph classes

  Does Geelen’s simulation conjecture hold for every nonempty proper class \(\mathcal C\subsetneq\mathcal G_{\mathrm{fin}}\) of finite simple graphs that is closed under isomorphism, local complementation, and vertex deletion?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Strong converse for the quantum capacity of a general channel

  Does every finite-dimensional memoryless quantum channel obey the strong converse at its unassisted quantum capacity?

  • Quantum Communication
  • Quantum capacity
  • Strong converses
  Solved
• Holevo random-coding exponent for classical–quantum channels

  Does every memoryless classical–quantum channel with a finite input alphabet and finite-dimensional output satisfy the random-coding lower bound in Eq. (5), including channels with mixed, noncommuting output states?

  • Quantum Communication
  • Classical capacity
  • Quantum coding theory
  Solved
• All-code exponential strong converse for degradable channels

  Does every finite-dimensional degradable quantum channel satisfy an all-code exponential strong converse for quantum communication at its quantum capacity?

  • Quantum Communication
  • Quantum capacity
  • Channel degradability
  • Strong converses
  Solved
• Exponential strong converse for transpose-degradable channels

  Does every finite-dimensional transpose-degradable channel satisfy an exponential strong converse for quantum communication at its single-letter quantum capacity?

  • Quantum Communication
  • Quantum capacity
  • Channel degradability
  • Strong converses
  Solved
• Secret-key capacity of thermal attenuators and amplifiers

  What is the secret-key capacity of a thermal attenuator for every transmissivity and thermal noise level?

  • Quantum Cryptography
  • Secret-key distillation
  • Bosonic channels
  Unsolved
• Bounded witnesses for every continuous-variable entangled state

  Does every entangled bosonic state admit a bounded entanglement witness?

  • Quantum Resource Theory
  • Quantum separability
  Solved
• LOCC entanglement cost of PPT-entangled Gaussian states

  What is the exact LOCC entanglement cost of a finite-energy PPT-entangled Gaussian state?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Bound entanglement
  • Entanglement cost
  Unsolved
• Additivity of entanglement of formation for two-mode Gaussian states

  Is unrestricted entanglement of formation additive on tensor powers of every finite-energy two-mode Gaussian state?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Entanglement cost
  • Additivity and regularization
  Unsolved
• Gaussian optimality of two-mode entanglement of formation

  Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every finite-energy two-mode Gaussian state?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Entanglement measures
  Solved
• Visible quantum compression with free shared randomness at the Holevo rate

  Does free shared classical randomness make the Holevo information achievable for visible compression of every finite mixed-state ensemble?

  • Quantum Communication
  • Quantum source coding
  • Channel simulation
  Unsolved
• Deterministic quadratic-size vertex-minor-universal graphs

  Does there exist an absolute constant \(C>0\) and a deterministic algorithm that, for each integer \(k\geq2\) supplied in unary, runs in time \(k^{O(1)}\) and outputs a \(k\)-vertex-minor-universal graph with at most \(Ck^2\) vertices?

  • Quantum Resource Theory
  • Quantum Algorithm
  • Quantum state preparation
  • Resource conversion
  • Computational complexity and computability
  Unsolved
• Loss of mixed unitarity after a positive time in a quantum dynamical semigroup

  Does there exist a norm-continuous semigroup \((\Phi_t)_{t\geq0}\) of unital completely positive trace-preserving maps on \(\mathcal L(\mathbb C^d)\), for some finite integer \(d\geq3\), such that \(\Phi_s\) is mixed unitary and \(\Phi_t\) is not mixed unitary for some \(0<s<t\)?

  • Quantum Communication
  • Quantum channel structure
  Unsolved
• Exact LOCC entanglement cost of the elegant joint measurement

  Is the exact deterministic LOCC entanglement cost of the two-qubit elegant joint measurement \(\mathcal M_{\mathrm{EJM}}\) equal to one ebit when arbitrary finite-dimensional pure entangled resources are allowed?

  • Quantum Resource Theory
  • Entanglement cost
  • Local operations and classical communication
  Unsolved
• Minimum output entropy additivity for qubit channels

  Is minimum output von Neumann entropy additive whenever one channel is a qubit channel?

  • Quantum Communication
  • Additivity and regularization
  • Matrix and entropy inequalities
  • Quantum channel structure
  Unsolved
• Permanent-of-Gaussians Conjecture

  Is the following Gaussian permanent estimation task \(\#\mathrm P\)-hard under randomized polynomial-time oracle reductions?

  • Quantum Algorithm
  • Computational complexity and computability
  • Quantum supremacy
  • Boson sampling
  Unsolved
• Gaussian entanglement of formation beyond bisymmetry

  Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every non-bisymmetric multimode Gaussian state?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Entanglement measures
  Unsolved
• Quantum capacity of a bosonic thermal attenuator

  What is the unassisted quantum capacity of a single-mode thermal attenuator at every strictly positive finite environment temperature?

  • Quantum Communication
  • Quantum capacity
  • Bosonic channels
  Unsolved
• Maximal length of stabilizer quantum MDS codes

  Is there a stabilizer quantum maximum-distance-separable code \([[n,k,d]]_q\) with \(k\geq1\) and \(d\geq3\) whose length \(n\) exceeds the conjectured limit \(b(q,d)\) of Eq. (2)?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Existence of a ((7,3,3)) qubit code

  Does a qubit quantum error-correcting code with parameters \(((7,3,3))_2\) exist, that is, a three-dimensional subspace of seven qubits with minimum distance at least three?

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Ordinary-Petz recovery bound for conditional mutual information

  Does the ordinary, unrotated Petz map universally recover a tripartite state with fidelity controlled by its conditional mutual information?

  • Quantum Communication
  • Quantum recovery
  • Matrix and entropy inequalities
  Solved
• Multi-slot overhead of virtual channel conjugation

  What is the optimal quasiprobability overhead of implementing the complex conjugate of an unknown quantum channel from \(n\) queries?

  • Quantum Algorithm
  • Quantum Resource Theory
  • Channel simulation
  • Superchannels and quantum combs
  • Quantum error mitigation
  Unsolved
• Closed-form exact PPT distillable entanglement

  What computable expression, if any, equals the regularized exact PPT distillable entanglement of a bipartite state?

  • Quantum Resource Theory
  • Entanglement distillation
  • PPT-preserving operations
  • Additivity and regularization
  • One-shot and finite-blocklength bounds
  Unsolved
• Amortization collapse for superchannel divergences

  Does amortization collapse for the geometric Rényi divergence of arbitrary finite-dimensional quantum superchannels?

  • Quantum Metrology
  • Quantum Communication
  • Channel discrimination
  • Quantum relative entropy
  • Superchannels and quantum combs
  Unsolved
• Asymptotic metrology with quantum-controlled causal order

  Does quantum control of causal order yield a persistent asymptotic metrological advantage over parallel access for some regular, finite-dimensional channel family?

  • Quantum Metrology
  • Quantum Resource Theory
  • Quantum estimation
  • Indefinite causal order
  Solved
• Nontrivial mutually degradable channel pairs

  Does there exist an integer \(d\geq2\) and a pair of distinct channels \(\mathcal M,\mathcal N:\mathcal L(A)\to\mathcal L(B)\), with \(A\simeq B\simeq\mathbb C^d\), that both have Choi rank exactly \(d\), are mutually degradable, and are each nondegradable?

  • Quantum Communication
  • Channel degradability
  • Quantum channel structure
  Solved
• Quantum LDPC codes at the Pauli hashing bound

  For which probability vectors \(\mathbf p=(p_I,p_X,p_Y,p_Z)\) can quantum low-density parity-check (LDPC) stabilizer codes reliably attain the exact hashing rate over the Pauli channel?

  • Quantum Error Correction
  • Quantum LDPC codes
  • Quantum coding theory
  • Decoding algorithms
  Unsolved
• Achievability of the Rains bound under completely PPT-preserving channels

  Is the Rains bound asymptotically achievable by completely positive-partial-transpose-preserving (completely PPT-preserving) channels for every full-rank two-qubit Bell-diagonal state, and what channel family achieves it?

  • Quantum Resource Theory
  • Entanglement distillation
  • PPT-preserving operations
  • Bell-diagonal states
  Unsolved
• Two-way quantum capacity: amplitude-damping channel

  What is the two-way quantum capacity \(\mathcal{Q}_2(\mathcal A_p)\) of the qubit amplitude-damping channel

  • Quantum Communication
  • Quantum capacity
  • Local operations and classical communication
  Unsolved
• Three-dimensional self-correcting quantum memory

  Does there exist a passive self-correcting quantum memory in three spatial dimensions?

  • Quantum Error Correction
  • Self-correcting quantum memories
  • Quantum thermodynamics
  • Quantum coding theory
  Unsolved
• Low-energy Holevo additivity for a squeezed thermal attenuator

  Is energy-constrained Holevo information additive for a squeezed thermal attenuator below its water-filling threshold?

  • Quantum Communication
  • Classical capacity
  • Bosonic channels
  • Additivity and regularization
  Unsolved
• Universality of arbitrary self-adjoint polynomial Hamiltonians

  Can every physical bosonic unitary be approximated by a single self-adjoint polynomial Hamiltonian evolution at any finite input energy?

  • Quantum Algorithm
  • Quantum circuit complexity
  • Hamiltonian simulation
  Solved
• Anticoncentration of independent complex Gaussian hafnians

  Do independent complex Gaussian hafnians satisfy a polynomial lower-tail bound at their root-mean-square scale?

  • Quantum Algorithm
  • Boson sampling
  • Computational complexity and computability
  Unsolved
• Universality of every fixed non-Gaussian polynomial generator

  Does every fixed non-Gaussian polynomial Hamiltonian become universal when all Gaussian controls are available?

  • Quantum Algorithm
  • Quantum circuit complexity
  • Hamiltonian simulation
  Unsolved
• Compact-time validity of the rotating wave approximation

  Does the Jaynes–Cummings evolution approximate the Rabi evolution strongly and uniformly on each compact time interval in the high-frequency limit?

  • Quantum Algorithm
  • Hamiltonian simulation
  Solved
• Two-way distillable entanglement of mixed two-mode Gaussian states

  What is the exact two-way distillable entanglement of a finite-energy mixed two-mode Gaussian state?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Entanglement distillation
  Unsolved
• Optimal success probability for Gaussian processing of non-Gaussian entanglement

  What is the maximal success probability for preparing an approximate two-mode squeezed vacuum from finitely many copies of a non-Gaussian state?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Entanglement distillation
  • One-shot and finite-blocklength bounds
  Unsolved
• Two-way quantum capacity of a thermal amplifier

  What is the two-way quantum capacity of a thermal amplifier for every gain and thermal noise level?

  • Quantum Communication
  • Quantum capacity
  • Bosonic channels
  Unsolved
• Gaussian-input preservation under a Gaussian reference extension

  Does preservation of Gaussian inputs by a trace-decreasing operation imply preservation when a Gaussian reference is attached?

  • Quantum Communication
  • Gaussian quantum information
  • Quantum channel structure
  Solved
• Strict stationary squeezing limit for one-mode thermal diffusion

  Does every stable one-mode thermal Gaussian diffusion obey the sharp stationary squeezing bound \(\lambda_{\min}(V_\infty)>\nu/2\)?

  • Quantum Resource Theory
  • Gaussian quantum information
  Solved
• Single-seed post-processing of Gaussian POVM densities

  Is every POVM with Gaussian density a classical post-processing of one displaced Gaussian seed?

  • Quantum Metrology
  • Gaussian quantum information
  Solved
• Gaussian-measurement equality with the Holevo bound

  When does the optimal single-copy Gaussian-measurement cost equal the Holevo bound?

  • Quantum Metrology
  • Quantum estimation
  • Gaussian quantum information
  Unsolved
• Average-case additive hardness of squared complex Gaussian hafnians

  Is additive estimation of squared complex Gaussian hafnians \(\#\mathrm P\)-hard under randomized polynomial-time oracle reductions?

  • Quantum Algorithm
  • Boson sampling
  • Computational complexity and computability
  Unsolved
• Unassisted quantum capacity of a thermal amplifier

  What is the unassisted quantum capacity of a thermal amplifier for every gain and thermal noise level?

  • Quantum Communication
  • Quantum capacity
  • Bosonic channels
  Unsolved
• Optimal output monitoring for parametric-oscillator squeezing

  Can any causal output measurement improve the mean conditional position squeezing achieved by ideal homodyne detection?

  • Quantum Resource Theory
  • Gaussian quantum information
  Solved
• Entropy photon-number inequality

  Does the entropy photon-number inequality hold for every pair of independent finite-energy bosonic inputs?

  • Quantum Communication
  • Bosonic channels
  • Matrix and entropy inequalities
  Unsolved
• Permanent anticoncentration for complex Gaussian matrices

  Do complex Gaussian permanents satisfy a polynomial lower-tail bound at their root-mean-square scale?

  • Quantum Algorithm
  • Boson sampling
  • Computational complexity and computability
  Solved
• Fully polynomial sampling of boson sampling with constant photon transmission

  For every fixed rational transmission \(0<\eta<1\), is boson sampling with independent photon loss classically samplable in fully polynomial time?

  • Quantum Algorithm
  • Boson sampling
  • Computational complexity and computability
  Unsolved
• Asymptotic nonclassical-state conversion by linear optics

  What is the optimal asymptotic conversion rate between finite-energy bosonic states under passive linear-optical protocols?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Resource conversion
  Unsolved
• Exact covariance criterion for bipartite Gaussian separability

  What necessary and sufficient condition on a covariance matrix characterizes bipartite Gaussian separability?

  • Quantum Resource Theory
  • Gaussian quantum information
  • Quantum separability
  Solved
• Polynomial-time quantum algorithm for Graph Isomorphism

  Does the Graph Isomorphism problem admit a polynomial-time quantum algorithm?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Optimal precision dependence of low-energy Hamiltonian simulation

  What is the tight precision dependence of worst-case query complexity for low-energy simulation in the regime (2)?

  • Quantum Algorithm
  • Hamiltonian simulation
  • Computational complexity and computability
  Unsolved
• Simultaneously optimal queries to both quantum linear-system oracles

  Can one quantum linear-system algorithm attain both query bounds in (2) simultaneously?

  • Quantum Algorithm
  • Quantum linear algebra
  • Computational complexity and computability
  Unsolved
• LOCC discrimination of three maximally entangled states

  Can every set of three orthogonal maximally entangled pure states on \(\mathbb C^d\otimes\mathbb C^d\), for every integer \(d\geq4\), be distinguished exactly from one copy using unrestricted two-way local operations and classical communication (LOCC)?

  • Quantum Metrology
  • Quantum Resource Theory
  • Quantum state discrimination
  • Local operations and classical communication
  Unsolved
• Exponential entanglement cost with simultaneous classical communication

  Does there exist a family of bipartite unitaries requiring exponentially many shared Bell pairs for fixed-error implementation when classical communication is limited to one simultaneous exchange?

  • Quantum Communication
  • Quantum Resource Theory
  • Entanglement cost
  • Quantum communication complexity
  • Local operations and classical communication
  Unsolved
• Quasi-linear graph-state resources for Pauli pairability

  Does there exist an absolute constant \(a\geq0\) such that \(N_{\mathrm P}(k)=O(k[\log_2(k+1)]^a)\) as the integer \(k\geq2\) tends to infinity?

  • Quantum Resource Theory
  • Quantum state preparation
  • Resource conversion
  Unsolved
• Second-level collapse of the exact PPT entanglement-cost hierarchy

  Does the semidefinite hierarchy for exact PPT entanglement cost collapse at its second level for every finite-dimensional bipartite density operator \(\rho_{AB}\)?

  • Quantum Resource Theory
  • Entanglement cost
  • PPT-preserving operations
  • Entanglement measures
  Unsolved
• Distributed compression of bipartite pure-state ensembles

  Determine the achievable rate region for independent local compression of an arbitrary finite ensemble of bipartite pure states.

  • Quantum Communication
  • Quantum source coding
  Unsolved
• Exact privacy-amplification exponent under relative-entropy security

  What is the optimal exponential rate of decay of relative-entropy insecurity achievable by two-universal hashing against quantum side information at every positive extraction rate below the conditional entropy?

  • Quantum Cryptography
  • Secret-key distillation
  • Quantum relative entropy
  Unsolved
• Quantum capacity of higher-dimensional depolarizing channels

  What is the unassisted quantum capacity of a depolarizing channel in every finite dimension \(d\geq3\)?

  • Quantum Communication
  • Quantum capacity
  • Additivity and regularization
  Unsolved
• Polynomial-time local-unitary equivalence of graph states

  Does a deterministic classical algorithm running in \(n^{O(1)}\) time decide local-unitary equivalence of arbitrary graph states on \(n\) labelled qubits?

  • Quantum Algorithm
  • Quantum Resource Theory
  • Local unitary equivalence
  • Computational complexity and computability
  Unsolved
• Chernoff exponents under separable and LOCC tests across copies

  Determine the optimal symmetric discrimination exponents for two faithful noncommuting states when measurements must be separable across copies or implemented by local operations and classical communication (LOCC) among the copies.

  • Quantum Metrology
  • Quantum Resource Theory
  • Quantum hypothesis testing
  • Local operations and classical communication
  • Quantum separability
  Unsolved
• Optimal random-unitary decomposition of symmetric Werner–Holevo channels

  For every odd integer \(d\geq5\), do there exist \(r=d(d+1)/2\) unitary operators \(U_1,\ldots,U_r\) on \(\mathbb C^d\) such that

  • Quantum Communication
  • Quantum channel structure
  Unsolved
• Quantum query complexity of Welded Tree path-finding

  Does finding an explicit ENTRANCE-to-EXIT path in the standard Welded Tree oracle problem require exponentially many quantum queries?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Two-way secret-key capacity of the qubit amplitude-damping channel

  What is the exact secret-key capacity \(K(\mathcal A_p)\) of the qubit amplitude-damping channel for every damping probability \(p\in[0,1]\)?

  • Quantum Cryptography
  • Quantum Communication
  • Secret-key distillation
  • Local operations and classical communication
  Unsolved
• Qubit code distance bound

  Does there exist a sequence of qubit quantum codes \(Q_j\subseteq(\mathbb C^2)^{\otimes n_j}\), with \(n_j\to\infty\), \(\dim Q_j\geq2\), and minimum distances \(d_j\), satisfying

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Polynomial-time quantum algorithm for approximate Shortest Vector Problem

  Does the polynomial-factor approximate Shortest Vector Problem admit a polynomial-time quantum algorithm?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Polynomial-time quantum algorithm for the Dihedral Hidden Subgroup Problem

  Does the Dihedral Hidden Subgroup Problem admit a quantum algorithm whose running time is polynomial in the input length?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Squashed entanglement of the qubit depolarizing channel

  What is the squashed entanglement \(E_{\mathrm{sq}}(\Lambda_p)\) of the qubit depolarizing channel, defined for a mixing parameter \(p\in[0,1]\) by

  • Quantum Resource Theory
  • Quantum Communication
  • Entanglement measures
  • Quantum capacity
  • Private capacity
  Unsolved
• Private capacity of the qubit depolarizing channel

  What is the private classical capacity \(P(\Lambda_p)\) of the qubit depolarizing channel, defined for a mixing parameter \(p\in[0,1]\) by

  • Quantum Communication
  • Quantum Cryptography
  • Private capacity
  • Additivity and regularization
  Unsolved
• Additivity of the relative entropy of entanglement

  Does the relative entropy of entanglement of every bipartite state equal its regularization, or is regularization genuinely necessary?

  • Quantum Resource Theory
  • Entanglement measures
  • Additivity and regularization
  • Quantum relative entropy
  Solved
• QMA(2) versus QMA

  Is every promise problem verifiable by a quantum Merlin-Arthur protocol with two unentangled witnesses also verifiable by a protocol with a single arbitrary witness, that is, is \(\mathsf{QMA}(2) = \mathsf{QMA}\) in the standard, unrelativized setting?

  • Quantum Algorithm
  • Computational complexity and computability
  • Quantum separability
  Unsolved
• Additivity of the entanglement of purification

  Is the entanglement of purification additive on tensor products?

  • Quantum Resource Theory
  • Entanglement measures
  • Additivity and regularization
  Unsolved
• Smallest output dimension violating minimum output entropy additivity

  What is the smallest output dimension in which the minimum output von Neumann entropy of quantum channels fails to be additive?

  • Quantum Communication
  • Additivity and regularization
  • Matrix and entropy inequalities
  • Quantum channel structure
  Unsolved
• Closed-form nonadditivity of the Holevo capacity and entanglement of formation

  Construct a simple closed-form or practically computable counterexample to additivity of the Holevo capacity or the entanglement of formation, together with a rigorous certificate of a strict violation.

  • Quantum Communication
  • Quantum Resource Theory
  • Classical capacity
  • Additivity and regularization
  • Entanglement measures
  Unsolved
• Computability of ordinary quantum capacity

  Is the ordinary unassisted quantum capacity a computable function of a finite description of a finite-dimensional quantum channel?

  • Quantum Communication
  • Quantum Algorithm
  • Quantum capacity
  • Computational complexity and computability
  Unsolved
• Parallel versus nested-adaptive superchannel discrimination

  Can nested-adaptive strategies improve the Stein exponent for discriminating two finite-dimensional quantum superchannels?

  • Quantum Metrology
  • Channel discrimination
  • Quantum hypothesis testing
  • Superchannels and quantum combs
  Unsolved
• Ordinary-Petz fidelity remainder for relative-entropy data processing

  Does the ordinary Petz recovery map give a universal fidelity remainder for monotonicity of quantum relative entropy?

  • Quantum Communication
  • Quantum recovery
  • Quantum relative entropy
  • Matrix and entropy inequalities
  Solved
• Entanglement of formation of generalized Bell-diagonal states

  For every local dimension \(d\geq3\), determine the entanglement of formation of an arbitrary Weyl–Bell-diagonal state.

  • Quantum Resource Theory
  • Entanglement measures
  • Bell-diagonal states
  Unsolved
• Minimal dimensions for strict transpose degradability

  What are the componentwise-minimal dimension triples \((d_A,d_B,d_E)\) that admit a transpose-degradable but nondegradable channel?

  • Quantum Communication
  • Channel degradability
  • Quantum channel structure
  Unsolved
• Positivity threshold for thermal-attenuator quantum capacity

  For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be

  • Quantum Communication
  • Quantum capacity
  • Bosonic channels
  Unsolved
• Unclassified existence parameters for homogeneous AME states

  For which of the parameter pairs specified below does an absolutely maximally entangled state exist?

  • Quantum Resource Theory
  • Quantum Error Correction
  • Absolutely maximally entangled states
  • Quantum coding theory
  Unsolved
• Weyl–Heisenberg-covariant SICs in every dimension

  Does every finite dimension admit a symmetric informationally complete measurement that is a single Weyl–Heisenberg orbit?

  • Quantum Metrology
  • Symmetric informationally complete measurements
  Unsolved
• Entanglement cost of an amplitude-damping-channel Choi state

  What is the entanglement cost of the Choi state of the qubit amplitude-damping channel

  • Quantum Resource Theory
  • Entanglement cost
  • Local operations and classical communication
  Unsolved
• Statistical strength of CGLMP measurements

  For every \(d\geq3\), do the standard CGLMP Fourier–phase measurements maximize the relative-entropy statistical strength against local realism among all projective \(d\)-outcome measurements on the fixed state \(\lvert\Phi_d\rangle=d^{-1/2}\sum_{j=0}^{d-1}\lvert j,j\rangle\), when the setting distribution is also optimized?

  • Quantum Resource Theory
  • Quantum Metrology
  • Bell nonlocality
  • Quantum hypothesis testing
  Unsolved
• Entanglement cost of a qubit Bell-diagonal state

  What is the entanglement cost of a qubit Bell-diagonal state for an arbitrary probability vector \(\mathbf p=(p_I,p_X,p_Y,p_Z)\)?

  • Quantum Resource Theory
  • Entanglement cost
  • Bell-diagonal states
  • Local operations and classical communication
  Unsolved
• Energy-constrained quantum capacity of a thermal attenuator

  For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be

  • Quantum Communication
  • Quantum capacity
  • Bosonic channels
  Unsolved
• Zauner-symmetric Weyl–Heisenberg SIC fiducials

  Does every finite dimension admit a Weyl–Heisenberg SIC fiducial that is an eigenvector of a Zauner Clifford unitary?

  • Quantum Metrology
  • Symmetric informationally complete measurements
  Unsolved
• Strict inclusion of degradable channels in the less-noisy class

  Do there exist finite-dimensional quantum channels that are less noisy in Watanabe’s regularized sense but are not degradable?

  • Quantum Communication
  • Channel degradability
  • Private capacity
  Solved
• Polynomial-time quantum algorithm for Learning With Errors

  Does the Learning With Errors problem in its standard worst-case-hard parameter regime admit a polynomial-time quantum algorithm?

  • Quantum Algorithm
  • Computational complexity and computability
  Unsolved
• Parity is not in QAC0

  Can polynomial-size constant-depth \(\mathsf{QAC}^0\) circuits compute the parity function, or equivalently, is \(\mathrm{PARITY}\notin\mathsf{QAC}^0\)?

  • Quantum Algorithm
  • Quantum circuit complexity
  Unsolved
• The PPT-squared conjecture

  Must the composition of any two compatible PPT completely positive maps be entanglement breaking?

  • Quantum Resource Theory
  • Quantum Communication
  • Quantum separability
  • Quantum channel structure
  Unsolved
• Is bipartite Quantum Max-Cut in BPP?

  Is the following bipartite Quantum Max-Cut promise problem in \(\mathrm{BPP}\)?

  • Quantum Algorithm
  • Computational complexity and computability
  • Hamiltonian complexity
  • Quantum Max-Cut
  Unsolved
• Average-case approximation hardness of random Ising partition functions

  Is it \(\#\mathrm{P}\)-hard to approximate \(|Z_R|^2\) to relative multiplicative error \(a+o(1)\) on a \(b\) fraction of random Ising instances?

  • Quantum Algorithm
  • Computational complexity and computability
  • Quantum supremacy
  • IQP sampling
  Unsolved
• Average-case approximation hardness of squared normalized gaps of random cubic polynomials

  Is it \(\#\mathrm{P}\)-hard to approximate \(\operatorname{ngap}(f)^2\) to relative multiplicative error \(a+o(1)\) on a \(b\) fraction of uniformly random degree-3 polynomials over \(\mathbb{F}_2\)?

  • Quantum Algorithm
  • Computational complexity and computability
  • Quantum supremacy
  • IQP sampling
  Unsolved
• Bidirectional classical-communication cost of bipartite channel simulation

  What is the asymptotic classical-communication cost of simulating a bipartite quantum channel with bidirectional classical communication and non-signalling assistance?

  • Quantum Communication
  • Quantum Resource Theory
  • Channel simulation
  • Quantum communication complexity
  • Quantum relative entropy
  • One-shot and finite-blocklength bounds
  Unsolved
• Universal purification with classically simulable operations

  Can classically simulable operations purify an unknown depolarized pure state from any number of copies?

  • Quantum Resource Theory
  • Quantum Algorithm
  • Quantum magic
  • Resource conversion
  • Quantum state preparation
  Unsolved
• Advantage of fully general superchannel-discrimination strategies

  Can a fully general adaptive strategy attain a larger Stein exponent than every nested-adaptive strategy for discriminating two quantum superchannels?

  • Quantum Metrology
  • Channel discrimination
  • Quantum hypothesis testing
  • Superchannels and quantum combs
  Unsolved
• Uniformly efficient HSW pretty-good decoding

  Can Holevo–Schumacher–Westmoreland codebooks operating at every rate below their ensemble Holevo information be chosen so that their square-root, or pretty-good, measurements have uniform quantum implementations whose cost is polynomial in the blocklength and…

  • Quantum Algorithm
  • Quantum Communication
  • Decoding algorithms
  • Classical capacity
  • Quantum state discrimination
  Unsolved
• Multimode constrained output entropy of a pure-loss channel

  Is Guha’s multimode Strong Conjecture 2 true for every correlated input state?

  • Quantum Communication
  • Bosonic channels
  • Matrix and entropy inequalities
  Unsolved
• Universal finite truncation of quantum and private capacities

  Do there exist channel-independent finite integers \(m_Q\) and \(m_P\) that determine, respectively, the quantum capacity and the private classical capacity of every finite-dimensional quantum channel?

  • Quantum Communication
  • Quantum Cryptography
  • Quantum capacity
  • Private capacity
  • Additivity and regularization
  Solved
• Candidate pure-loss second-order converse

  Does the pure-loss bosonic channel admit the following candidate second-order classical converse under a maximum-photon-number occupation constraint?

  • Quantum Communication
  • Bosonic channels
  • Classical capacity
  • One-shot and finite-blocklength bounds
  Unsolved
• Exactly solvable nondegradable quantum channels

  Does there exist a finite-dimensional quantum channel that is neither degradable nor antidegradable and whose unassisted quantum capacity is known exactly?

  • Quantum Communication
  • Quantum capacity
  • Channel degradability
  Solved
• Structural criterion for classical–entanglement trade-off advantage

  Characterize the finite-dimensional quantum channels for which joint classical–entanglement coding strictly outperforms time sharing between unassisted and unlimited-entanglement classical communication.

  • Quantum Communication
  • Quantum Resource Theory
  • Classical capacity
  • Entanglement-assisted communication
  • Additivity and regularization
  Unsolved
• Strong converse for general mixed-state quantum compression

  Does general finite-dimensional i.i.d.

  • Quantum Communication
  • Quantum source coding
  • Strong converses
  Unsolved
• Quantum capacity of the Gaussian random-displacement channel

  What is the unconstrained, unassisted quantum capacity of the single-mode Gaussian random-displacement channel?

  • Quantum Communication
  • Quantum capacity
  • Bosonic channels
  Unsolved
• Convergence of the JRF iteration for mixed-state discrimination

  Does the Ježek–Řeháček–Fiurášek (JRF) iteration, initialized by the uniform POVM, converge to a globally optimal minimum-error measurement for every finite ensemble containing mixed quantum states?

  • Quantum Metrology
  • Quantum state discrimination
  Unsolved
• Minimum LU–LC counterexample for graph states

  What is the least number of qubits for which two graph states can be locally unitary equivalent without being locally Clifford equivalent?

  • Quantum Resource Theory
  • Local unitary equivalence
  Solved
• Equal-weight low-Choi-rank decompositions of quantum channels

  Can every finite-dimensional quantum channel be written as the uniform mixture of \(d_B\) channels whose Choi ranks are at most the input dimension?

  • Quantum Communication
  • Quantum channel structure
  Unsolved
• Umegaki relative entropy of local recovery

  Does the conditional mutual information of every finite-dimensional tripartite state dominate its Umegaki relative entropy of local recovery?

  • Quantum Communication
  • Quantum recovery
  • Quantum relative entropy
  • Matrix and entropy inequalities
  Solved
• Multiplicativity for polarized Werner–Holevo channels

  For every integer \(d\geq3\), every \(x\in(0,1)\), and every \(1<p<2\), is the maximal output Schatten \(p\)-norm of the polarized Werner–Holevo channel multiplicative on two identical copies?

  • Quantum Communication
  • Additivity and regularization
  • Quantum channel structure
  • Matrix and entropy inequalities
  Unsolved
• The quantum PCP conjecture

  Is the constant-relative-gap local Hamiltonian problem QMA-hard?

  • Quantum Algorithm
  • Hamiltonian complexity
  • Computational complexity and computability
  Unsolved
• Delayed-onset additivity violation for minimum output Rényi entropy

  Does there exist a finite-dimensional quantum channel whose minimum output Rényi entropy is additive for every tensor power below some order and first becomes strictly subadditive at that order?

  • Quantum Communication
  • Additivity and regularization
  • Matrix and entropy inequalities
  • Quantum channel structure
  Unsolved
• Transpose degradability beyond degradability

  Does there exist a finite-dimensional transpose-degradable quantum channel that is not degradable?

  • Quantum Communication
  • Channel degradability
  • Quantum channel structure
  Unsolved
• Finite nontrivial LU moduli of AME states

  Do there exist integers \(N,d\geq 2\) for which the absolutely maximally entangled states of \(N\) qudits of local dimension \(d\) form finitely many, but more than one, local-unitary equivalence classes?

  • Quantum Resource Theory
  • Absolutely maximally entangled states
  • Local unitary equivalence
  Unsolved
• Generalized Stein lemma for fully quantum channel resources

  Let \(\mathcal N:\mathcal L(A)\to\mathcal L(B)\) be a finite-dimensional quantum channel.

  • Quantum Resource Theory
  • Quantum Metrology
  • Quantum hypothesis testing
  • Channel discrimination
  • Quantum relative entropy
  Unsolved
• POVM steering threshold of higher-dimensional Werner states

  What is the exact steering threshold for arbitrary POVMs on a higher-dimensional Werner state?

  • Quantum Resource Theory
  • Quantum steering
  Unsolved
• Secret key from every Bell-nonlocal behavior

  Does every finite-alphabet Bell-nonlocal behavior have a strictly positive asymptotic secret-key rate against arbitrary individual nonsignalling attacks?

  • Quantum Cryptography
  • Quantum Resource Theory
  • Secret-key distillation
  • Device-independent cryptography
  • Bell nonlocality
  Unsolved
• SIC-POVM existence in every dimension

  Does a symmetric informationally complete positive-operator-valued measure exist in every finite dimension \(d\ge2\)?

  • Quantum Metrology
  • Symmetric informationally complete measurements
  Unsolved
• Unconditional classical verification with one quantum prover

  Does every language \(L\in\mathsf{BQP}\) admit a single-prover interactive proof with a fully classical verifier, an efficient quantum honest prover, and information-theoretic soundness?

  • Quantum Algorithm
  • Verification of quantum computation
  • Computational complexity and computability
  Unsolved
• Finite-alphabet nonsignalling simulation of entangled qubits

  For every partially entangled two-qubit pure state, does some fixed finite-alphabet nonsignalling resource give an exact noncommunicating simulation of all local projective measurements?

  • Quantum Resource Theory
  • Bell nonlocality
  • Resource conversion
  Unsolved
• Quantum capacity of a qubit Pauli channel

  What is the quantum capacity \(\mathcal{Q}(\Lambda_{\mathbf p})\) of the qubit Pauli channel defined by

  • Quantum Communication
  • Quantum capacity
  Unsolved
• Universal simulation with two PR boxes

  Can shared randomness and at most two Popescu–Rohrlich boxes exactly simulate every pair of local projective measurements on every two-qubit state, without communication?

  • Quantum Resource Theory
  • Bell nonlocality
  • Resource conversion
  Unsolved
• Resources for implementing Gibbs-preserving channels

  What are the exact one-shot coherence and work costs of implementing an arbitrary Gibbs-preserving channel by thermal operations?

  • Quantum Resource Theory
  • Quantum thermodynamics
  • Channel simulation
  • One-shot and finite-blocklength bounds
  • Resource conversion
  Unsolved
• Collective cost of tensor-power state preparation

  Determine the asymptotic weighted circuit cost of preparing tensor powers of a known pure state, and characterize when collective preparation is cheaper per copy than independent preparation.

  • Quantum Algorithm
  • Quantum state preparation
  • Quantum circuit complexity
  • Additivity and regularization
  Unsolved
• One-bit simulation of partially entangled qubits

  Can shared randomness and one classical bit exactly simulate every pair of local projective measurements on every pure entangled two-qubit state?

  • Quantum Communication
  • Quantum Resource Theory
  • Bell nonlocality
  • Quantum communication complexity
  Unsolved
• Long-range vacuum CHSH violation

  Does the vacuum of a massive free scalar Bose field violate the CHSH inequality between bounded localization regions at arbitrarily large spacelike separation?

  • Quantum Resource Theory
  • Bell nonlocality
  Unsolved
• Trace-exponential lower bound for matrix-word averages

  Is the normalized trace average of all words in two positive-definite matrices always bounded below by the corresponding trace exponential?

  • Quantum Communication
  • Matrix and entropy inequalities
  Unsolved
• NPT bound entanglement and the rank-five frontier

  Does there exist a finite-dimensional bipartite state with non-positive partial transpose (NPT) that is undistillable by local operations and classical communication (LOCC)?

  • Quantum Resource Theory
  • Entanglement distillation
  • Bound entanglement
  • Local operations and classical communication
  Unsolved
• Minimal-support frontier for absolutely maximally entangled states

  Determine, for every integer \(d\geq 2\), the largest number \(\mathcal N(d)\) of \(d\)-level parties that admit an absolutely maximally entangled state of minimal computational-basis support.

  • Quantum Resource Theory
  • Quantum Error Correction
  • Absolutely maximally entangled states
  • Quantum coding theory
  Unsolved
• Lockability of two-way distillable entanglement

  Can discarding one local qubit reduce two-way distillable entanglement by an arbitrarily large amount?

  • Quantum Resource Theory
  • Entanglement distillation
  • Entanglement measures
  • Local operations and classical communication
  Unsolved
• Polynomial shared-resource lower bounds for routing

  Does an explicit total Boolean family require polynomial shared-state cost for bounded-error one-round \(f\)-routing?

  • Quantum Communication
  • Quantum Cryptography
  • Quantum communication complexity
  • Position-based quantum cryptography
  Unsolved
• Extensible causality and process purification

  Is every finite-dimensional extensibly causal process matrix purifiable?

  • Quantum Resource Theory
  • Indefinite causal order
  Unsolved
• Absolute separability from spectra

  Characterize the spectra of bipartite states that remain separable under every global unitary, and decide whether absolute separability equals absolute positivity under partial transpose in higher local dimensions.

  • Quantum Resource Theory
  • Quantum separability
  Unsolved
• Tough error models

  Determine \(c(e,n)\) and construct tough error models in the sense of [KW05].

  • Quantum Error Correction
  • Quantum coding theory
  Unsolved
• Semialgebraicity of closed quantum correlations

  Is the closure of the finite-dimensional tensor-product correlation set semialgebraic in every fixed bipartite Bell scenario?

  • Quantum Resource Theory
  • Bell nonlocality
  Unsolved
• Distillable entanglement of Bell-diagonal states

  What is the distillable entanglement \(D(\rho_{\mathbf p})\) under local operations and classical communication (LOCC) for the Bell-diagonal state

  • Quantum Resource Theory
  • Entanglement distillation
  • Bell-diagonal states
  • Local operations and classical communication
  Unsolved
• Secret key from every entangled state

  Does every finite-dimensional entangled bipartite state have positive asymptotic distillable secret key?

  • Quantum Cryptography
  • Quantum Resource Theory
  • Secret-key distillation
  • Bound entanglement
  • Local operations and classical communication
  Unsolved
• Maximum number of mutually unbiased bases

  For each integer \(d\geq2\), determine the maximum number \(\mu(d)\) of pairwise mutually unbiased orthonormal bases of \(\mathbb C^d\).

  • Quantum Metrology
  • Mutually unbiased bases
  Unsolved
• Optimal CGLMP measurements for a maximally entangled state

  For every \(d\geq3\), do the standard Fourier–phase measurements optimize the CGLMP violation of the maximally entangled state among all projective \(d\)-outcome measurements?

  • Quantum Resource Theory
  • Quantum Metrology
  • Bell nonlocality
  Unsolved
• Quantum violations of bipartite Bell facets

  Must every nontrivial facet Bell inequality in a finite bipartite scenario have a quantum violation?

  • Quantum Resource Theory
  • Bell nonlocality
  Unsolved
• Relative entropy of entanglement for two qubits

  Find a closed formula for the relative entropy of entanglement of every two-qubit density operator \(\rho\), including an explicit closest separable state.

  • Quantum Resource Theory
  • Entanglement measures
  • Quantum separability
  • Quantum relative entropy
  Unsolved
• Square-root remainder in generalized quantum equipartition

  Let \(H\) be a \(d\)-dimensional Hilbert space.

  • Quantum Resource Theory
  • Quantum Communication
  • Quantum hypothesis testing
  • Quantum relative entropy
  • One-shot and finite-blocklength bounds
  Unsolved
• Complete facet descriptions for Bell polytopes

  Determine complete facet descriptions for local-behavior polytopes beyond the presently solved Bell scenarios, either for a specified unresolved finite scenario or for a nontrivial infinite family with additional structure.

  • Quantum Resource Theory
  • Bell nonlocality
  Unsolved
• Constant trace-distance separability testing

  What is the computational complexity of testing bipartite separability with a constant trace-distance promise gap?

  • Quantum Algorithm
  • Quantum Resource Theory
  • Quantum separability
  • Computational complexity and computability
  Unsolved
• Honest-party lockability of distillable key

  Can loss of one qubit held by an honest party reduce the two-way distillable secret key by an arbitrarily large amount?

  • Quantum Cryptography
  • Quantum Resource Theory
  • Secret-key distillation
  • Local operations and classical communication
  Unsolved
• Real four-quhex absolutely maximally entangled state

  Does there exist an \(\operatorname{AME}(4,6)\) state whose coefficients are real in a product basis?

  • Quantum Resource Theory
  • Absolutely maximally entangled states
  Unsolved
• Capacity-achieving codes for amplitude damping

  What is the constructive quantum code for achieving the quantum capacity \(\mathcal{Q}(\mathcal A_p)\) of the qubit amplitude-damping channel

  • Quantum Error Correction
  • Quantum Communication
  • Quantum capacity
  • Quantum coding theory
  • Decoding algorithms
  Solved