Top 100 Quantum Theorems

M. David, J. Palsberg

This list collects 100 important theorems in quantum physics, quantum information and quantum computing. Inspired by the list of the top 100 mathematical theorems of which Freek Wiedijk tracks the formalization progress, this is a similarly arbitrary and necessarily incomplete list with no particular order.

We welcome submissions from your interactive theorem proving community, and will link formal statements and proofs below.

Acknowledgements. Thank you to Rodolfo R. Soldati, Alex Meiburg, Serge Massar, Tein van der Lugt, Debbie Leung, Matilde Baroni, Eduardo Martín-Martínez, Jean-Yves Desaules, Maksym Serbyn for their contributions. We are grateful for support by the NSF Challenge Institute for Quantum Computation.

The List

1. The Spectral Theorem
2. Eigenstates in an infinite square well
3. The Quantum Harmonic Oscillator
4. Ehrenfest’s Theorem
5. Quantum tunneling through a finite potential barrier
6. Conservation of the probability (4-)current
7. Eigenstates of a δ-distribution well
8. Heisenberg Uncertainty Relation
9. Baker–Campbell–Hausdorff Formula
10. Trotter Product Formula (Lie–Trotter–Kato)
11. Simultaneous diagonalization of commuting observables
12. Noether’s Theorem
13. Wigner’s Theorem
14. Stone–von Neumann Theorem
15. Spectrum of the hydrogen atom (Coulomb potential)
16. Addition of angular momentum (e.g. Clebsch–Gordan)
17. Wigner–Eckart Theorem
18. Power-series spectrum of a perturbed time-independent Hamiltonian
19. Dyson series for time-dependent perturbation theory
20. Correctness of the Rayleigh–Ritz method
21. Correctness of the Born–Oppenheimer approximation
22. The Adiabatic Theorem
23. Berry phase (Geometric phase)
24. Superdense coding
25. Quantum teleportation with shared entanglement
26. No-Cloning Theorem / No-Broadcast Theorem / No-Teleportation Theorem
27. No-Communication Theorem
28. No-Deleting Theorem
29. Schrödinger–HJW Theorem (purification of mixed states)
30. Schmidt decomposition for Hilbert spaces
31. Bell’s Theorem (QM violates CHSH inequality)
32. Monogamy of entanglement
33. Tsirelson bound
34. Correctness of the Deutsch–Jozsa algorithm
35. Correctness of Shor’s algorithm
36. Correctness of Grover’s algorithm
37. The Bennett–Bernstein–Brassard–Vazirani Theorem (BBBV)
38. Quantum key distribution by BB84
39. Impossibility of quantum bit commitment (Mayers–Lo–Chau)
40. Solovay–Kitaev Theorem
41. Universality of sets of two-qubit quantum gates
42. Universality of the Toffoli and the Hadamard
43. Five two-qubit gates are necessary and sufficient to implement a Toffoli gate
44. Six two-qubit linear nearest-neighbor gates are necessary and sufficient to implement a Toffoli gate
45. 3n CX gates are necessary to implement a multi-control Toffoli gate with n-1 controls
46. Adiabatic and gate-based quantum computing are equivalent up to polynomial factors
47. The Threshold Theorem (quantum fault-tolerance)
48. Petz recovery map
49. Choi’s Theorem on completely positive maps
50. Choi–Jamiołkowski isomorphism (channel-state duality)
51. Gelfand–Naimark–Segal construction
52. Gelfand–Naimark Theorem
53. Krein–Milman Theorem
54. Stinespring’s Factorization Theorem / Naimark’s Dilation Theorem
55. Continuous Functional Calculus
56. Gleason’s Theorem
57. Holevo’s Theorem
58. Pusey–Barrett–Rudolph Theorem
59. Kochen–Specker Theorem
60. The Spectral Theorem for PVMs
61. Strong subadditivity of quantum entropy
62. Entanglement-assisted classical capacity of quantum channels
63. Quantum state discrimination for two states (Ivanović–Dieks–Peres Limit)
64. The Lloyd–Shor–Devetak Theorem for quantum channel capacity
65. Eastin–Knill Theorem
66. Gottesman–Knill Theorem
67. Magic state distillation
68. Correctness of distillation of Bell pairs
69. 1D gapped Hamiltonians have area-law ground states
70. The Lieb–Robinson bound
71. Onsager’s solution to the 2D Ising model
72. MIP* = RE
73. BQP \(\subseteq\) PP
74. QIP = PSPACE
75. Quantum Stein Lemma
76. Quantum 3-SAT is QMA1-complete
77. Computing the Jones polynomial at roots of unity is BQP-hard
78. Uniqueness of 4-dimensional representations of the Clifford algebra (up to unitary equivalence)
79. Spin-statistics Theorem
80. Reeh–Schlieder Theorem
81. Wick’s Theorem
82. Elitzur’s Theorem
83. 2D TQFTs are equivalent to Frobenius algebras
84. Chern–Simons theory can compute the Jones polynomial
85. The CFT central charge is its entanglement entropy
86. Osterwalder–Schrader Reconstruction Theorem
87. 4D Gaussian free field theory satisfies the Osterwalder-Schrader axioms
88. Bisognano–Wichmann Theorem
89. Wood–Spekkens–Bell Theorem
90. Correctness of block encoding
91. Correctness of Hamiltonian simulation by Trotterization
92. Correctness of Hamiltonian simulation by Linear Combination of Unitaries
93. Correctness of Hamiltonian simulation by Qubitization
94. Qubit reuse is NP-complete
95. Correctness of a quantum circuit optimizer
96. Correctness of the toric code
97. Correctness of entanglement-assisted stabilizer codes
98. The Knill–Laflamme conditions
99. Correctness of approximate circuit synthesis
100. Correctness of Simon’s algorithm