Jordan Cotler Harvard Society of Fellows 210104634 Aharonov Cotler Qi Introduction what is an experiment Measuring constants of nature speed of light charge of electron Quantifying dynamical properties rate of a chemical reaction ID: 1012693
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1. Quantum Algorithmic MeasurementJordan CotlerHarvard Society of Fellows2101.04634 [Aharonov, Cotler, Qi]
2. Introduction: what is an experiment? Measuring constants of nature (speed of light, charge of electron) Quantifying dynamical properties (rate of a chemical reaction) Inferring structural properties (symmetry group of a crystal) Learning more abstract information (the chain of chemical reactions athat comprise photosynthesis, whether Yang-Mills describes the astrong force)
3. Introduction: what is an experiment? Measuring constants of nature (speed of light, charge of electron) Quantifying dynamical properties (rate of a chemical reaction) Inferring structural properties (symmetry group of a crystal) Learning more abstract information (the chain of chemical reactions athat comprise photosynthesis, whether Yang-Mills describes the astrong force)What exactly is an experiment, in its full scope of generality?
4. Quantum Extended Church-Turing Thesis
5. Quantum Extended Church-Turing ThesisA universal quantum computer can efficiently simulate any physical system
6. Q: Is an experiment a type of algorithm?Q: What is the complexity of an experiment?Q: Can tools from quantum algorithms speed up experiments?Q: Is there a provable advantage of coherent vs. incoherent apparatus?
7. Q: Is an experiment a type of algorithm?Q: What is the complexity of an experiment?Q: Can tools from quantum algorithms speed up experiments?Q: Is there a provable advantage of coherent vs. incoherent apparatus?We will address these questions as well as establish a new kind of exponential advantage for quantum experiments
8. Would like to abstract the notion of an experiment
9. Would like to abstract the notion of an experiment
10. Would like to abstract the notion of an experiment
11. Would like to abstract the notion of an experiment Interaction between nature and labInteraction between lab and working space
12. Would like to abstract the notion of an experiment Interaction between nature and labInteraction between lab and working spaceAn experiment is a hybrid of a black box algorithm and interactive protocol
13. Would like to abstract the notion of an experiment Interaction between nature and labInteraction between lab and working spaceAn experiment is a hybrid of a black box algorithm and interactive protocol
14. Would like to abstract the notion of an experiment Interaction between nature and labInteraction between lab and working spaceAn experiment is a hybrid of a black box algorithm and interactive protocol
15. Would like to abstract the notion of an experiment Interaction between nature and labInteraction between lab and working spaceAn experiment is a hybrid of a black box algorithm and interactive protocol
16. Would like to abstract the notion of an experiment Interaction between nature and labInteraction between lab and working spaceAn experiment is a hybrid of a black box algorithm and interactive protocol
17. Would like to abstract the notion of an experiment An experiment is a hybrid of a black box algorithm and interactive protocol
18. Would like to abstract the notion of an experiment InputsOutputsAn experiment is a hybrid of a black box algorithm and interactive protocol
19. Would like to abstract the notion of an experiment InputsOutputsThree definitions: Lab oracle, QUALM, TaskAn experiment is a hybrid of a black box algorithm and interactive protocol
20. Would like to abstract the notion of an experiment Lab oracleInputsOutputsThree definitions: Lab oracle, QUALM, TaskAn experiment is a hybrid of a black box algorithm and interactive protocol
21. Would like to abstract the notion of an experiment Lab oracleQuantum Algorithmic MeasurementInputsOutputsThree definitions: Lab oracle, QUALM, TaskAn experiment is a hybrid of a black box algorithm and interactive protocol
22. Would like to abstract the notion of an experiment QUALM(Lab Oracle, Inputs) Outputs Lab oracleQuantum Algorithmic MeasurementInputsOutputs
23. Would like to abstract the notion of an experiment QUALM(Lab Oracle, Inputs) Outputs Lab oracleQuantum Algorithmic MeasurementInputsOutputsA Task is a problem specification, i.e. specifies what we want the QUALM to do
24. InputsOutputsExample 1: X-ray diffraction
25. InputsOutputsExample 1: X-ray diffraction
26. OutputsExample 1: X-ray diffraction
27. OutputsExample 1: X-ray diffraction
28. Example 1: X-ray diffractionLab oracleOutputs
29. Example 1: X-ray diffractionLab oracleLab oracles: , , … Outputs
30. Example 1: X-ray diffractionLab oracleLab oracles: , , … Outputs
31. Example 1: X-ray diffractionLab oracleLab oracles: , , … OutputsContains:detectors,laser, laptop,…
32. Example 1: X-ray diffractionLab oracleLab oracles: , , … Task is to find shape of a crystalOutputsContains:detectors,laser, laptop,…
33. Example 1: X-ray diffractionLab oracleLab oracles: , , … Task is to find shape of a crystalX-ray diffraction is QUALM achieving the TaskOutputsContains:detectors,laser, laptop,…
34. Example 2: Distinguishing two statesInputsOutputs
35. Example 2: Distinguishing two statesInputsOutputs✓
36. Example 2: Distinguishing two statesOutputs
37. Example 2: Distinguishing two statesOutputs
38. Example 2: Distinguishing two statesOutputsLab oracle
39. Example 2: Distinguishing two statesOutputsTwo possible lab oracles: Lab oracle
40. Example 2: Distinguishing two statesOutputsTwo possible lab oracles: and Lab oracle
41. Example 2: Distinguishing two statesOutputsTwo possible lab oracles: and Task is to find a QUALM satisfying: QUALM() = 0, QUALM() = 1 Lab oracle
42. Example 3: Quantum algorithmsInputsOutputs
43. Example 3: Quantum algorithmsInputsOutputs
44. Other examples:
45. Other examples: We believe that all known and possible experiments fit into aaaaaaaaaaaaaaathis paradigm
46. Other examples: We believe that all known and possible experiments fit into aaaaaaaaaaaaaaathis paradigm Quantum algorithms, quantum state tomography, quantum Hamiltonian tomography, quantum verification protocols, quantum process tomography, quantum cooling protocols, quantum machine learning, quantum communication protocols, Gibbs sampling, quantum imaging, quantum control theory,…
47. Other examples: We believe that all known and possible experiments fit into aaaaaaaaaaaaaaathis paradigm Quantum algorithms, quantum state tomography, quantum Hamiltonian tomography, quantum verification protocols, quantum process tomography, quantum cooling protocols, quantum machine learning, quantum communication protocols, Gibbs sampling, quantum imaging, quantum control theory,… We will see that the QUALM framework helps us to analyze experiments using tools from quantum complexity theory and to ask new questions; the generality of the framework inspires new and novel experiments
48. A quantum circuit point of viewWe write out the circuit in spacetime and can see the application of the lab oracle which couples Nature and the lab, as well as the quantum circuits which couple the lab and the working spaceData of the lab oracle includes the superoperator and the initial state of Nature Data of the QUALM includes the quantum circuits acting jointly on the lab and working space
49. Complexity of a QUALM
50. Complexity of a QUALM
51. Complexity of a QUALMA combination of computational complexity and query complexity Our analyses will combine techniques from each type of complexity theory
52. Complexity of a TaskThe complexity of a Task is the minimum complexity over QUALMs achieving that TaskQuantifies the minimal resources required to achieve the Task
53. Types of accessNLWHow is information communicated between the lab and working space?
54. Types of accessNLWCoherent access: Information communicated by quantum channelIncoherent access: Information communicated by classical channel a(i.e., via LOCC’s) Lab system completely measured between lab aoracle accessesHow is information communicated between the lab and working space?
55. Types of accessCoherent access: Information communicated by quantum channelIncoherent access: Information communicated by classical channel a(i.e., via LOCC’s) Lab system completely measured between lab aoracle accessesHow is information communicated between the lab and working space?Quantum memoryQuantum computation on stored quantum dataNo quantum memoryQuantum computation on stored classical dataCan still employ adaptive strategies
56. Complexity of a Task with specified access type
57. Other access types
58. Other access typesQUALM complexity advantage is physically important!
59. Other access types[Huang, Kueng, Preskill ‘21][Bacon, Childs, van Dam ‘05]QUALM complexity advantage is physically important!
60. Other access typesQUALM complexity advantage is physically important![Huang, Kueng, Preskill ‘21][Bacon, Childs, van Dam ‘05]
61. A comment on Simon’s algorithmSimon’s algorithm sits within the non-adaptive, incoherent access settingThe quantum algorithm only requires the experimentalist to prepare product states and make product measurementsAccordingly, Simon’s algorithm provides an exponential advantage of non-adaptive, incoherent access QUALMS over classical QUALMsBy contrast, our work is concerned with an exponential advantage of coherent access QUALMs over adaptive, incoherent access QUALMs
62. QUALM hierarchyClassicalNon-adaptive, incoherent accessAdaptive, incoherent accessCoherent access
63. QUALM hierarchyClassicalNon-adaptive, incoherent accessAdaptive, incoherent accessCoherent accessExponential advantage due to Simon’s algorithmExponential advantage due to [Aharonov, Cotler, Qi ‘21]
64. Fixed unitary problemIntuition: It is difficult to distinguish time-independent evolution from time-aaaaaaaaidependent evolution if each is sufficiently chaoticWe will make this intuition precise by first formulating a Task.
65. Fixed unitary problemInputsOutputs
66. InputsOutputs✓Fixed unitary problem
67. OutputsFixed unitary problem
68. OutputsFixed unitary problem
69. OutputsFixed unitary problem
70. OutputsLab oracleFixed unitary problem
71. OutputsTwo lab oracles: and Lab oracleFixed unitary problem
72. OutputsTwo lab oracles: and Lab oracleFixed unitary problemi.i.d. Haar-randomHaar-random, same each time
73. OutputsTwo lab oracles: and Task is to find a QUALM satisfying: QUALM() = 0, QUALM() = 1 Lab oracleFixed unitary problemi.i.d. Haar-randomHaar-random, same each time
74. Fixed unitary problem
75. Fixed unitary problemTheorem [Aharonov, Cotler, Qi ‘21]
76. Fixed unitary problemTheorem [Aharonov, Cotler, Qi ‘21]Establishes an exponential separation between coherent access and incoherent access A new kind of exponential separation in quantum complexity theory
77. Fixed unitary problemTheorem [Aharonov, Cotler, Qi ‘21]Linear upper bound on coherent access task complexity: Easy Just find a single coherent access QUALM that worksExponential lower bound on incoherent access task complexity: Hard Need to rule out all efficient incoherent access QUALMs (!!)
78. Fixed unitary problemLower bounding incoherent access QUALMs1. Reduce to analyzing “simple measurement aaiQUALMS” (single measurement and single aaipreparation between each lab oracle call)Sketch of proof:2. Show that even for adaptive simple aaimeasurement QUALMs, the final output aaicannot distinguish from with aai. lab oracle calls
79. Fixed unitary problemLower bounding incoherent access QUALMs1. Reduce to analyzing “simple measurement aaiQUALMS” (single measurement and single aaipreparation between each lab oracle call)Sketch of proof:2. Show that even for adaptive simple aaimeasurement QUALMs, the final output aaicannot distinguish from with aai. lab oracle calls Uses techniques from representation theory, combinatorics, and random matrix theory (particularly Weingarten functions)
80. Symmetry distinction problem
81. Symmetry distinction problemIntuition: It is difficult to determine the symmetry classes of a strongly-aaaaaaaiiiinteracting quantum many-body systemWe formulate a Task in which this can be made precise.
82. Symmetry distinction problemIntuition: It is difficult to determine the symmetry classes of a strongly-aaaaaaaiiiinteracting quantum many-body systemWe formulate a Task in which this can be made preciseFirst, recall time-reversal symmetry classes of unitaries:UnitaryOrthogonalSymplectic
83. Symmetry distinction problemOutputs
84. Symmetry distinction problemOutputs
85. OutputsLab oracleSymmetry distinction problem
86. Lab oracles: Lab oracleSymmetry distinction problemOutputs
87. Lab oracles: Lab oracleSymmetry distinction problemOutputs Haar-random, same each time
88. Lab oracles: Task to find QUALM satisfying: QUALM() = 0, QUALM() = 1, QUALM() = 2 Lab oracleSymmetry distinction problemOutputs Haar-random, same each time
89. Theorem [Aharonov, Cotler, Qi ‘21]Symmetry distinction problem
90. Theorem [Aharonov, Cotler, Qi ‘21]Linear upper bound on coherent access task complexity: Easy (but interesting) Find a single coherent access QUALM that worksExponential lower bound on incoherent access task complexity: Hard Rule out all efficient incoherent access QUALMsSymmetry distinction problem
91. DiscussionQuantum supremacyQuantum computers better than classical computers at classical problems
92. DiscussionQuantum supremacyQuantum computers better than classical computers at classical problemsQUALM supremacyQuantum coherence has advantage over incoherence for quantum experiments
93. Discussion We have established a new framework for analyzing quantum experiments using quantum complexity theory; opens up a whole new set of questions We have established that coherent coupling between experimental apparatus and a quantum computer can lead to exponential savings in resources Experimental demonstration of QUALM advantage in the NISQ era? What are effects of noise? Other, more physical examples with coherent QUALM advantage?