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Umesh V. Vazirani U. C. Berkeley
Challenges in Quantum Information Science
1st 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.
1st 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 2nd quantum revolution: Synthesize new quantum systems
i.e. Quantum devices.
Superposition Principle
+ -
ψ =α 0 +β 1
α2+ β
2=1
Qubit:
Superposition Principle
+ -
ψ =α 0 +β 1
α2+ β
2=1
Qubit:
Measure: outcome = 0 with probability
outcome = 1 with probability
α2
β2
Hilbert space is Large!
n particles
State=Ψ = αx xx∑ αx
x∑
2=1
all n-bit strings
Hilbert space is Large!
n particles
State=Ψ = αx xx∑ =
α0000
α0001
.
.α1111
#
$
%%%%%%%
&
'
(((((((all n-bit
strings
Quantum computation teaches us that quantum systems are exponentially complex:
Classical: O(n) parameters.
Quantum: 2O(n) parameters. Exponential power of quantum computers.
n particles
Unitary Evolution
€
1 0 0 00 1 0 00 0 0 i0 0 −i 0
#
$
% % % %
&
'
( ( ( (
⊗ In−2
all n-bit strings
Ψ = αx xx∑ =
α0000
α0001
.
.α1111
#
$
%%%%%%%
&
'
(((((((
Unitary Evolution
€
1 0 0 00 1 0 00 0 0 i0 0 −i 0
#
$
% % % %
&
'
( ( ( (
⊗ In−2
all n-bit strings
Ψ = α 'x xx∑ =
α '0000α '0001..
α '1111
#
$
%%%%%%%
&
'
(((((((
Limited Access - Measurement
input
output
∑=Ψx
x xα 1|| 2=∑x
xα
§ Measurement: See with probability |αx|2
€
x
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. ”
• 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
Impact of Quantum Computers
• 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:
Testing quantum devices poses fundamental new challenges:
• exponential complexity:
Classical: O(n) parameters.
Quantum: 2O(n) parameters. • Also exponentially private! Holevo: Can access at most O(n) parameters
n particles
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 =1200 +
1211
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”.
Clauser Horn Shimoni Holt 1969: Simplified “test for quantumness”. Aspect 1981: experimental test. Hensen et al, Nature Oct 2015: Loophole-free
€
ψ =1200 +
1211
CHSH Game
Input: x εR {0,1} Output: a ε {0,1}
Input: y εR {0,1} Output: b ε {0,1}
Maximize Pr[xy = ] Classically it is impossible to do better than 0.75 If DA and DB share entangled qubits, then they can achieve success probability cos2 π/8 ≈ 0.85 Violation of Bell Inequality.
a⊕ b
Test of Quantumness
Quantum Strategy for CHSH Game:
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
€
ψ =1200 +
1211
Bell Basis States
€
ψ =1200 +
1211
€
0
€
1
€
0
€
1
Measurement reveals same outcome on both qubits
Bell Basis States
ψ =1200 +
1211
= 12uu +
12u⊥u⊥
Rotational Invariance: Always see matching outcomes €
0
€
1
€
0
€
1
€
u€
u⊥
€
u€
u⊥
Bell Basis States
Probability of matching outcomes = cos2 θ
Probability of different outcomes = sin2 θ
€
0
€
1
€
0
€
1
€
u€
u⊥
€
v
€
v⊥
θ
€
uψ =
1200 +
1211
= 12uu +
12u⊥u⊥
θ
θ vs sin2θ
CHSH Game
Input: x εR {0,1} Output: a ε {0,1}
Input: y εR {0,1} Output: b ε {0,1}
Maximize Pr[xy = ] Classically it is impossible to do better than 0.75 If DA and DB share entangled qubits, then they can achieve success probability cos2 π/8 ≈ 0.85 Violation of Bell Inequality.
a⊕ b
Quantum Strategy for CHSH Game:
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
€
ψ =1200 +
1211
Quantum Key Distribution
• Feature: Unconditional security No computational assumptions • BB84: Prepare and measure • Proof of unconditional security: [Mayers ‘01], [Shor&Preskill ’00]
0, 1
K K
0 , + ,...
• Goal: Establish secure shared random key between distant users.
1200 +
1211
Assume that adversary Eve manufactured quantum boxes, and can share entanglement with them.
[Myers & Yao ’98] DIQKD Challenge: quantum devices completely untrusted. Test that they behave as claimed.
Beyond unconditional security
1200 +
1211
Assume that adversary Eve manufactured quantum boxes, and can share entanglement with them.
[Myers & Yao ’98] DIQKD Challenge: quantum devices completely untrusted. Test that they behave as claimed.
Beyond unconditional security
[Ekert 91] Protocol based on testing Bell pairs
Proof of fully device independent QKD. Constant key rate while tolerating constant noise rate.
[V, Vidick PRL 2014]
1200 +
1211
Inputs Alice: 0,1,2 Bob: 0,1 On input 0,1 Perform Bell test Alice input = 2: measure as Bob on input 1.
Proof of fully device independent QKD. Constant key rate while tolerating constant noise rate.
[V, Vidick PRL 2014]
1200 +
1211
Inputs Alice: 0,1,2 Bob: 0,1 On input 0,1 Perform Bell test Alice input = 2: measure as Bob on input 1.
Monogamy test
Key generation
1200 +
1211
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…
Security against Quantum Adversary
110100010111…
ε-close to uniform distribution in total variation distance.
Certifiable Quantum Generator
log n log 1/ε truly random bits
n bits
[Colbeck Phd thesis ‘09] Pironio, et al. Nature 464, 1021-1024 (15 April 2010) [V. Vidick STOC 2012]
110100010111…
Certifiable Quantum Generator log n log 1/ε truly random bits
n bits
Certifies that this particular output string is random!! And fresh!
A B
x1 … xn y1 … yn
a1 … an b1 … bn
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]
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] [Fitzsimons, Kashefi 2013] [Aharonov, Ben-Or, Eban, Mahadev 2013]
Mildly Quantum Verifier – crypto approach
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
• 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.
Summary
A pragmatic approach to testing Special purpose quantum computers
Quantum annealers
min − Jiji~ j∑ σ iσ j
s
σ ∈ −1,1{ }
• Finding the lowest energy state of a classical Hamiltonian. (e.g. Ising spin glass) • These are NP-complete CSPs (constraint satisfaction problems)!
Quantum Annealing
min − Jiji~ j∑ σ iσ j
s
σ ∈ −1,1{ }
1
Supplementary material for “Quantum annealing with more than one hundred qubits”
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1
FIG. 1: Qubits and couplers in the D-Wave device.The D-Wave One Rainer chip consists of 4 ⇥ 4 unit cells ofeight qubits, connected by programmable inductive couplersas shown by lines.
I. OVERVIEW
Here we provide additional details in support of themain text. Section II shows details of the chimera graphused in our study and the choice of graphs for our simula-tions. Section III expands upon the algorithms employedin our study. Section IV presents additional success prob-ability histograms for di↵erent numbers of qubits and forinstances with magnetic fields, explains the origin of easyand hard instances, and explains how the final state canbe improved via a simple error reduction scheme. SectionV presents further correlation plots and provide moredetails on gauge averaging. Section VI gives details onhow we determined the scaling plots and how quantumspeedup can be detected on future devices. Finally, sec-tion VII explains how the spectral gaps were calculatedby quantum Monte Carlo (QMC) simulations.
II. THE CHIMERA GRAPH OF THE D-WAVEDEVICE.
The qubits and couplers in the D-Wave device can bethought of as the vertices and edges, respectively, of abipartite graph, called the “chimera graph”, as shown infigure 1. This graph is built from unit cells containingeight qubits each. Within each unit cell the qubits andcouplers realise a complete bipartite graph K4,4 whereeach of the four qubits on the left is coupled to all ofthe four on the right and vice versa. Each qubit on theleft is furthermore coupled to the corresponding qubitin the unit cell above and below, while each of the oneson the right is horizontally coupled to the correspond-ing qubits in the unit cells to the left and right (withappropriate modifications for the boundary qubits). Ofthe 128 qubits in the device, the 108 working qubits usedin the experiments are shown in green, and the couplersbetween them are marked as black lines.
For our scaling analysis we follow the standard pro-cedure for scaling of finite dimensional models by con-sidering the chimera graph as an L ⇥ L square latticewith an eight-site unit cell and open boundary condi-tions. The sizes we typically used in our numerical sim-ulations are L = 1, . . . , 8 corresponding to N = 8L2 =8, 32, 72, 128, 200, 288, 392 or 512 spins. For the simu-lated annealers and exact solvers on sizes of 128 andabove we used a perfect chimera graph. For sizes below128 where we compare to the device we use the workingqubits within selections of L⇥L eight-site unit cells fromthe graph shown in figure 1.
In references [29, 33] it was shown that an optimi-sation problem on a complete graph with
pN vertices
can be mapped to an equivalent problem on a chimeragraph with N vertices through minor-embedding. Thetree width of
pN mentioned in the main text arises from
this mapping. See Section VIA for additional detailsabout the tree width and tree decomposition of a graph.
III. CLASSICAL ALGORITHMS
A. Simulated annealing
Simulated annealing (SA) is performed by using theMetropolis algorithm to sequentially update one spin af-ter the other. One pass through all spins is called onesweep, and the number of sweeps is our measure of theannealing time for SA. Our highly optimised simulated
Quantum Annealing: • Start with x-field: qubits in state
• Gradually turn on z-z coupling between qubits, while turning down x-field. • Final Hamiltonian
• System at finite temperature
120 +
121
H f = − Jiji~ j∑ σ i
zσ jz
H0 = σ ix
i∑
• 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:
A classical benchmark for quantum annealers [Shin, Smith, Smolin, V]
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.
