1. New quantum error-correcting codes for a bosonic mode
AQIS Conference 2016
Taipei, Taiwan
Most of the talk is about arXiv: (PRX)
Marios H. Michael
Barbara M. Terhal
Matti Silveri
S. M. Girvin
R. T. Brierley
Liang Jiang
Victor V. Albert
Philip Reinhold
Juha Salmilehto
Kyungjoo Noh

2.   Why do we want to encode in an oscillator?⊗
⊗ … ⊗ 
Experimentally, microwave cavities have longer (ms) lifetimes than (related) qubits.
Reagor, … Schoelkopf, APL2013, PRB2016
States in microwave cavities can be controlled.
Heeres, … Schoelkopf, PRL2015
Dominant error channels for oscillators are simpler than those for multi-qubit paradigm.
E.g., less ancillas, measurements

3. Why do we want to consider new codes?
Cavities/fibers aren’t perfect, have errors:
Photon loss 𝑎 (with [ 𝑎 , 𝑎  ]=1)
Dephasing errors 𝑛 ≡ 𝑎  𝑎
Photon gain errors a 
“Standard” encodings (polarization, angular momentum, occupation number) do not protect from errors. Multimode encodings require more oscillators. But Hilbert space of one oscillator is already big. Why not utilize it?
We already have! Full QEC has already been done on a code we will consider (Ofek, … Schoelkopf, Nature 2016).

4. How do we encode in an oscillator?
A code {| 𝜇 code ⟩} is a subspace of the full oscillator Hilbert space (i.e., Fock space).
Logical states can be expressed in Fock state basis |𝑛⟩:
For example, the occupation number code states are

5. Codes that we consider:
GKP codes, e.g.,
Cat codes, e.g.,
Binomial codes (new), e.g.,
Numerical/optimized codes (ask computer what works)
Gottesman, Kitaev, Preskill, PRA 2001
Leghtas et al., PRL 2013
Mirrahimi, Leghtas, VVA et al., NJP 2014
Michael, Silveri, Brierley, VVA et al., PRX 2016

6. Experimental code progress report
as of AQIS 2016
Code
En/decoding
Gates
QEC
cat
 [1]
 [2]
 [3]
bin  
gkp
[1] Vlastakis et al., Science 2013
[2] Heeres et al., arXiv:
[3] Ofek et al., Nature 2016
 Currently under investigation
 Possible using techniques from Heeres et al., PRL 2015; Krastanov, VVA, et al., PRA.

7. Codes that we consider:
GKP codes, e.g.,
Cat codes, e.g.,
Binomial codes (new), e.g.,
Numerical/optimized codes (ask computer what works)
Gottesman, Kitaev, Preskill, PRA 2001
Leghtas et al., PRL 2013
Mirrahimi, Leghtas, VVA et al., NJP 2014
Michael, Silveri, Brierley, VVA et al., PRX 2016

8. 1. How binomial codes protect from loss
This particular binomial code protects from one lowering operator 𝑎 . Upon undergoing the error,
Quantum information is preserved! It just moved to the error subspace {|1⟩,|3⟩}, and can be moved back.
Since average occupation number of both states is 2, one loss event does not destroy the information.

9. 2. How binomial codes protect from no loss
Unfortunately, no loss does not imply no damage? For | 0 bin ⟩, the longer the system goes without a loss event 𝑎 , the more likely it is to be in |0⟩ and the less likely it is to be in |4⟩.
This effect --- the no-jump evolution --- is manifested by Kraus operator (with damping parameter 𝛾)
that decays all states to |0⟩.

10. 2. How binomial codes protect from no loss
Under the no-jump evolution ( 𝑛 =2 is avg. occ. num.):
| 1 bin ⟩ is not affected, but | 0 bin ⟩ is:
Within first order in 𝛾, one can rotate the information from |𝐸⟩ back to 0 bin without affecting | 1 bin ⟩.

11. Summary: protection from loss and no loss.
Both events can thus be corrected as follows:
If a loss event 𝑎 detected, apply rotation from {|1⟩,|3⟩} subspace to logical subspace.
While no loss detected, continuously apply 𝛾-dependent rotation from |𝐸⟩ to logical subspace.
Therefore, amplitude damping channel E + recovery R leave state invariant (within first order in 𝛾):

12. Why did we call them binomial codes?
Superimposing to write in the conjugate basis:
Coefficients are square roots of binomial coefficients; “ ” from Pascal’s triangle.
This yields a natural generalization…

13. Binomial codes: general case
General formula for binomial codes:
Spacing 𝑆>0 determines how many loss events 𝑎 𝑠≤𝑆 the code protects from.
Order 𝑁>0 determines how many moments of 𝑛 𝑛≤𝑁 are equal for both logical states.

14. Binomial codes: general case
General formula for binomial codes:
Codes are customizable! For example, picking 𝑁=𝑆, one can show that binomial code can protect from amplitude damping up to order 𝑂( 𝛾 𝑆 ):

15. So we have all these codes…
GKP codes, e.g.,
Cat codes, e.g.,
Binomial codes (new), e.g.,
Numerical/optimized codes (ask computer what works)
Gottesman, Kitaev, Preskill, PRA 2001
Leghtas et al., PRL 2013
Mirrahimi, Leghtas, VVA et al., NJP 2014
Michael, Silveri, Brierley, VVA et al., PRX 2016

16. …How do we compare them?
We can use entanglement fidelity 𝐹 𝑒 , optimizable using a semi-definite program (Fletcher, Shor, Win PRA2007).
VVA et al., in preparation

17. Conclusion
Continuous variables, even using only one mode, offer long lifetimes, a high degree of experimental control, and tractable error channels. QEC has already been achieved*!
Binomial codes can have lower mean occupation number 𝑛 than previous codes. They are also customizable to protect against any combinations of 𝑎 and 𝑎  .
There are now several codes:
It is a good time to benchmark them, both analytically and
numerically!
Reagor et al., APL2013, PRB2016
Heeres et al., arXiv:
Vlastakis et al., Science 2013
*Ofek et al., Nature 2016
Heeres et al., PRL2015
Krastanov, et al., PRA
Michael, Silveri, Brierley, VVA, Salmilehto, Jiang, Girvin PRX
gkp Gottesman, Kitaev, Preskill, PRA 2001
cat Leghtas et al., PRL 2013; Mirrahimi et al., NJP 2014
VVA et al., in preparation…