![Connection with boson sampling
Ground
Excited
state
Define a ‘vacuum state’ | ...
We can introduce ‘creation’ and ‘annihilation’ operators
.||ˆ|,|ˆ
iii i
cc
ˆ ˆ ˆ ˆ ˆ ˆ[ , ] 0, fori j i j j ic c c c c c i j
1
ˆ ˆ| | | 1 |a b a bc
](https://image.slidesharecdn.com/opticsseminar1-180905213518/85/What-is-the-point-of-Boson-sampling-14-320.jpg)
![Connection with boson sampling
Ground
Excited
state
Define a ‘vacuum state’ | ...
We can introduce ‘creation’ and ‘annihilation’ operators
.||ˆ|,|ˆ
iii i
cc
ˆ ˆ ˆ ˆ ˆ ˆ[ , ] 0, fori j i j j ic c c c c c i j
n
nmnm cc )0(ˆ)(ˆ
Time dynamics of state can be transferred to c operators. One can then show that
the between any two times, the c operators are related by a linear optics type
relation.
where is a unitary matrix.
1
ˆ ˆ| | | 1 |a b a bc
](https://image.slidesharecdn.com/opticsseminar1-180905213518/85/What-is-the-point-of-Boson-sampling-15-320.jpg)
What is the point of Boson sampling?
- 1. What is the point of boson sampling? Thomas Brougham Quantum Theory Group
- 2. Other work Can encode information in temporal degrees of freedom of photons. Two approaches : time-bins and temporal envelope create temporally entangled photons using SPDC We can combine both approaches to create photons that are entangled both in time-bins and temporal envelope!
- 3. Outline 1) Motivation for boson sampling 2) What is boson sampling 3) Cryptographic applications 4) Dynamics of spin Hamiltonians
- 4. Motivation Lots of computationally difficult problems, i.e. problems where the time to solve scales badly with the size of problem . A quantum computer could help! We can encode all information as string of 0 and 1s. Encode 0 and 1s directly on quantum system allows for the possibility of a quantum superposition, i.e. qubits. 1|0|| ba Lots of big claims about quantum computers (QC). Need experimental verification. Building universal QC is difficult, so instead build one that solves one problem.
- 6. Quantum Sampling 1| 100% reflecting mirror 1| 0| We have a linear optical network of beam-splitters, with inputs either single photons or vacuum. Detectors Sets of possible outcomes 2 0 0 1 1 0 0 2 0 1 0 1 0 0 2 0 1 1 Question: Can we determine the probability to obtain some a particular set of measurement outcomes?
- 7. Boson Sampling 1| 0| Now consider a large linear optical network of beam-splitters, described by some unitary U. Suppose the number of input modes is much greater than the number of photons. Detectors Question: what is the probability to obtain the various possible sets of outcomes? U Mathematically, this is equivalent to calculating the permanent of sub-matrices of U.
- 8. What is a permanent? Consider a square matrix 333231 232221 131211 aaa aaa aaa A 332211332112322311322113312312312213 333231 232221 131211 aaa+aaa-aaa-aaa+aaa+aaa )det( aaa aaa aaa A 332211332112322311322113312312312213 aaa+aaaaaaaaa+aaa+aaa )(Perm A
- 9. Boson Sampling 1| 0| Detectors Question: what is the probability to obtain the various possible sets of outcomes? U Mathematically, this is equivalent to calculating the permanent of sub-matrices of U. Number of steps in calculation scales exponentially with number of input modes and photons. But, a linear optical experiment could sample the probabilities. Experimental demonstration of boson sampling is proof of quantum advantage
- 10. Cryptography U Alice BobAlice and Bob share a secret n-bit string U ...1101|| input Alice and Bob independently determine the set of outcomes with maximum probability Can encode output onto a new bit string. lengthofstringnewashareBobandAlice D sets of possible outcomes D sets of possible outcomes nD )(log2
- 11. Spin dynamics , , ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ' 1 .d n n mn m n m n ij i j i j n m n i j m n i j H B Z K X X Y Y K X Y Y X 1 2 3 4 5 6 7 8 Example: d=8 Consider a network of coupled spin ½ particles.
- 12. Is simulating spins difficult? Suppose we had a network of d coupled spin-1/2 particles. At time t=0, Alice prepares N of nodes spin up, while the other d-N are prepared as spin down. Alice Bob ?)...( P Bob then tries to calculate the probability for sets of nodes to be spin up and down at time t=.
- 13. Is simulating spins difficult? Suppose we had a network of d coupled spin-1/2 particles. At time t=0, Alice prepares N of nodes spin up, while the other d-N are prepared as spin down. Alice Bob ?)...( P Bob then tries to calculate the probability for sets of nodes to be spin up and down at time t=. This probability is related to the permanent of matrix!
- 14. Connection with boson sampling Ground Excited state Define a ‘vacuum state’ | ... We can introduce ‘creation’ and ‘annihilation’ operators .||ˆ|,|ˆ iii i cc ˆ ˆ ˆ ˆ ˆ ˆ[ , ] 0, fori j i j j ic c c c c c i j 1 ˆ ˆ| | | 1 |a b a bc
- 15. Connection with boson sampling Ground Excited state Define a ‘vacuum state’ | ... We can introduce ‘creation’ and ‘annihilation’ operators .||ˆ|,|ˆ iii i cc ˆ ˆ ˆ ˆ ˆ ˆ[ , ] 0, fori j i j j ic c c c c c i j n nmnm cc )0(ˆ)(ˆ Time dynamics of state can be transferred to c operators. One can then show that the between any two times, the c operators are related by a linear optics type relation. where is a unitary matrix. 1 ˆ ˆ| | | 1 |a b a bc
- 16. Conclusions Boson sampling is interesting! Proves a possible experimental test of power of quantum computation But there can be other applications such as in cryptography and simulating spins Other applications
- 17. Thank you