Challenges in Quantum Information Science. Umesh V. Vazirani U. C. Berkeley

Challenges in Quantum Information Science Umesh V. Vazirani U. C. Berkeley

1 st quantum revolution - Understanding physical world: periodic table, chemical reactions electronic wavefunctions underlying semiconductor physics Model of Computers: Based on Mechanistic/Clockwork Universe Extended Church-Turing thesis: Any real world computer can be efficiently simulated on a Turing Machine. [Feynman 81, Bernstein, V 93] Quantum computers violate Extended Church-Turing thesis [Bennett, Brassard 84] Quantum key distribution 2 nd quantum revolution: Synthesize new quantum systems i.e. Quantum devices.

Superposition Principle Qubit: ψ = α 0 + β 1 + - α 2 + β 2 =1

Superposition Principle Qubit: ψ = α 0 + β 1 + - α 2 + β 2 =1 Measure: outcome = 0 with probability outcome = 1 with probability α 2 β 2

Hilbert space is Large! n particles State= Ψ = α x x α 2 x =1 all n-bit strings x x

Hilbert space is Large! n particles # % % % State= Ψ = α x x = % x % % $ all n-bit strings α 0000 α 0001.. α 1111 & ( ( ( ( ( ( '

Quantum computation teaches us that quantum systems are exponentially complex: Classical: O(n) parameters. n particles Quantum: 2 O(n) parameters. Exponential power of quantum computers.

Unitary Evolution # 1 0 0 0& % 0 1 0 0 ( % ( I % 0 0 0 i n 2 ( % $ 0 0 i 0 ( ' # % % % Ψ = α x x = % x % % $ all n-bit strings α 0000 α 0001.. α 1111 & ( ( ( ( ( ( '

Unitary Evolution # 1 0 0 0& % 0 1 0 0 ( % ( I % 0 0 0 i n 2 ( % $ 0 0 i 0 ( ' # % % % Ψ = α ' x x = % x % % $ all n-bit strings α ' 0000 α ' 0001.. α ' 1111 & ( ( ( ( ( ( '

Limited Access - Measurement input output Ψ = α x x α 2 x = 1 x x Measurement: See x with probability α x 2

Quantum computers do NOT provide a uniform speed up over classical computers. Only certain problems have the structure that permits speedup.

Quantum computers do NOT provide a uniform speed up over classical computers. Only certain problems have the structure that permits speedup. Shor s Algorithm - Efficient Factoring Breaks Elliptic curve cryptography (Some) Private key cryptography Grover s algorithm - Quadratic speedup of search (Some) Linear algebra, machine learning tasks Efficient simulation of quantum systems

Soon, my friends, you will look at a child's homework and see nothing to eat.

Impact of Quantum Computers Thomas Watson I think there is a world market for maybe 5 computers. Near term: Post-quantum cryptography Classical public-key cryptosystems that resist quantum cryptanalysis. NIST is currently creating standards based on lattice cryptosystems. Long term: Simulation of quantum systems and nanoscience

Tremendous recent progress and confidence among experimentalists about prospects for implementation of small to medium-scale quantum communication and computation devices

Tremendous recent progress in experimental realization of quantum communication and computation devices

Martinis Group: Linear array of 9 superconducting qubits Protection of classical states from bit flip errors

Monroe Group (UMD): Five qubit trapped-ion quantum computer. Gate fidelity 98-99% Deutsch-Jozsa 95% Bernstein-Vazirani 90%

Tremendous recent progress in experimental realization of quantum communication and computation devices But Devices unreliable Special purpose (limited control) Difficult to characterize precisely (full tomography impractical) Bringing together theory of untrusted quantum devices with experimental developments will be critical to further progress.

Tremendous recent progress and confidence among experimentalists about prospects for implementation of small to medium-scale quantum communication and computation devices But Devices unreliable Special purpose (limited control) Difficult to characterize precisely (full tomography impractical) Bringing together theory of untrusted quantum devices with experimental developments will be critical to further progress.

Testing quantum devices poses fundamental new challenges:

Testing quantum devices poses fundamental new challenges: exponential complexity: Classical: O(n) parameters. n particles Quantum: 2 O(n) parameters. Also exponentially private! Holevo: Can access at most O(n) parameters

Testing quantum devices poses fundamental new challenges: Emerging theory of quantum testing of: Quantum cryptographic devices Quantum key distribution Quantum randomness generation Quantum computers

Part II: Pragmatic approach to testing special purpose quantum computers, such as the D-Wave quantum annealer.

Test for quantumness EPR Paradox 1935: spooky action at a distance ψ = 1 2 00 + 1 2 11 Both particles give same outcome no matter what (basis) measurement is performed on them. This holds even if they are widely separated, e.g. they are in distant galaxies.

Test for quantumness EPR Paradox 1935: spooky action at a distance John Bell 1964: Entanglement gives rise to non-classical correlations. i.e. quantum mechanics is incompatible with local hidden variable theory. Test for quantumness. ψ = 1 2 00 + 1 2 11 Clauser Horn Shimoni Holt 1969: Simplified test for quantumness. Aspect 1981: experimental test. Hensen et al, Nature Oct 2015: Loophole-free

Test of Quantumness CHSH Game Input: x ε R {0,1} Output: a ε {0,1} Input: y ε R {0,1} Output: b ε {0,1} a b Maximize Pr[xy = ] Classically it is impossible to do better than 0.75 If D A and D B share entangled qubits, then they can achieve success probability cos 2 π/8 0.85 Violation of Bell Inequality.

Quantum Strategy for CHSH Game: ψ = 1 2 00 + 1 2 11 Input: x ε {0,1} Output: a ε {0,1} Input: y ε {0,1} Output: b ε {0,1} x and y random. Max Pr[xy = a+b (mod 2)] Alice: if x = 0, measure in standard basis x = 1, measure in π/4 basis Bob: if y = 0, measure in π/8 basis y = 1, measure in π/8 basis

Bell Basis States 1 1 0 ψ = 1 2 00 + 1 2 11 0 Measurement reveals same outcome on both qubits

Bell Basis States u 1 u 1 u ψ = 1 2 00 + 1 2 11 u 0 = 1 2 uu + 1 2 u u 0 Rotational Invariance: Always see matching outcomes

Bell Basis States u 1 1 u 0 ψ = 1 2 00 + 1 2 11 = 1 2 uu + 1 2 u u v θ v u 0 Probability of matching outcomes = cos 2 θ Probability of different outcomes = sin 2 θ

θ vs sin 2 θ θ

CHSH Game Input: x ε R {0,1} Output: a ε {0,1} Input: y ε R {0,1} Output: b ε {0,1} a b Maximize Pr[xy = ] Classically it is impossible to do better than 0.75 If D A and D B share entangled qubits, then they can achieve success probability cos 2 π/8 0.85 Violation of Bell Inequality.

Quantum Key Distribution Goal: Establish secure shared random key between distant users. 0, +,... 0, 1 K K Feature: Unconditional security No computational assumptions BB84: Prepare and measure Proof of unconditional security: [Mayers 01], [Shor&Preskill 00]

Beyond unconditional security 1 2 00 + 1 2 11 [Myers & Yao 98] DIQKD Challenge: quantum devices completely untrusted. Test that they behave as claimed. Assume that adversary Eve manufactured quantum boxes, and can share entanglement with them.

[V, Vidick PRL 2014] 1 2 00 + 1 2 11 Proof of fully device independent QKD. Constant key rate while tolerating constant noise rate. Inputs Alice: 0,1,2 Bob: 0,1 On input 0,1 Perform Bell test Alice input = 2: measure as Bob on input 1.

Security against Quantum Adversary 1 2 00 + 1 2 11 Assume that adversary Eve manufactured quantum boxes, and can share entanglement with them. Eve cannot guess shared random key è fresh randomness! Based on Monogamy of entanglement

Certifiable Quantum Generator log n log 1/ε truly random bits n bits 110100010111 ε-close to uniform distribution in total variation distance. [Colbeck Phd thesis 09] Pironio, et al. Nature 464, 1021-1024 (15 April 2010) [V. Vidick STOC 2012]

Certifiable Quantum Generator log n log 1/ε truly random bits n bits 110100010111 Certifies that this particular output string is random!! And fresh!

x 1 x n y 1 y n A B a 1 a n b 1 b n Output is certifiably random provided: Outputs pass a simple statistical test. No-signaling condition is satisfied e.g. based on speed of light limits imposed by relativity. In particular, convincing even to quantum skeptic!

Testing that a claimed quantum computer is really quantum classical channel Draws on Theory of interactive proof systems from computational complexity theory New properties of entanglement, encryption

Mildly Quantum Verifier crypto approach classical channel small quantum channel Arthur has constant # bits of quantum storage + quantum channel to Merlin. [Aharonov, Ben-Or, Eban 09] [Broadbent, Fitzsimons, Kashefi 09]

Testing that a claimed quantum computer is really quantum classical channel classical channel Reichardt, Unger, V. Nature 496, 456 460 (25 April 2013)

Testing that a claimed quantum computer is really quantum classical channel Major open question

Summary Quantum key distribution and quantum random number generation: - Very efficient tests - Next challenge is experimental realization. Testing of quantum computers: - Proof of concept but great challenges in making these robust and efficient - Major open question: purely classical verifier testing single quantum computer.

A pragmatic approach to testing Special purpose quantum computers Quantum annealers

Quantum Annealing Finding the lowest energy state of a classical Hamiltonian. (e.g. Ising spin glass) These are NP-complete CSPs (constraint satisfaction problems)! s min J ij σ i σ j i~ j σ { 1,1 }

s min J ij σ i σ j i~ j σ { 1,1 }

Quantum Annealing: 0 4 8 12 16 20 24 28 1 2 3 32 5 6 7 36 9 10 11 40 13 14 15 44 17 18 19 48 21 22 23 52 25 26 27 56 29 30 31 60 Start with x-field: qubits in state x 1 H 2 0 + 1 2 1 0 = σ i i 33 34 35 37 38 39 41 42 43 45 46 47 49 50 51 53 54 55 57 58 59 61 62 63 Gradually turn on z-z coupling between qubits, while turning down x-field. 64 65 68 69 72 73 76 77 80 81 84 85 88 89 92 93 Final Hamiltonian z H f = J ij σ iz σ j i~ j 66 67 70 71 74 75 78 79 82 83 86 87 90 91 94 95 System at finite temperature 96 100 104 108 112 116 120 124 97 101 105 109 113 117 121 125 98 102 106 110 114 118 122 126 99 103 107 111 115 119 123 127

A classical benchmark for quantum annealers [Shin, Smith, Smolin, V] Correct classical model to capture large scale algorithmic features quantum annealers is not simulated annealing, but a system of interacting magnets. Suitable noise model Quantum Turing Test: Classical model Quantum annealer

Conclusions Exciting time for quantum computing: Experimental breakthroughs and promise. Implications for foundations of QM Bringing together theory of untrusted quantum devices with experimental developments key to further progress. Testing quantum devices è tests of QM beyond Bell tests. Classical benchmarks and quantum Turing tests as a pragmatic approach to quantum testing.