Quantum’error’correc,on’and’ thefutureofsolidstate’ …...Stop measuring one Plaquette...
Transcript of Quantum’error’correc,on’and’ thefutureofsolidstate’ …...Stop measuring one Plaquette...
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Quantum error correc,on and the future of solid state qubits
David DiVincenzo 26.06.2012
Varenna Course CLXXXIII
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Basics of quantum error correc>on -‐ digi>zing quantum errors -‐ a discrete group theory, finding good codes -‐ circuits for error correc>on, fault tolerance
Codes for integrated circuits -‐ surface code -‐ the basic device experiments -‐ new tools for fault tolerance: making and braiding holes
Outline
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T dt 'exp Bi t '( )! Si t '( )i=1"#
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BI ! I +BX ! X +BY !Y +BZ ! Z
Two great things about this paper: 1) Made evident the fact (clarified by others) that quantum errors are discrete.
General con>nuous-‐>me Bath-‐System quantum evolu>on
For a one-‐qubit system:
X, Y, Z are Pauli matrices
If error correc>on procedure corrects for “bit flip” (X), “π-‐phase error” (Z), then it also corrects Y=iZX, and, by linearity, corrects the most general system-‐bath coupling.
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Two great things about this paper: 2) Found a code that corrects against single-‐qubit error.
Triple-‐repe>>on code inside itself.
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Quantum error correction implemented by quantum circuits
Classical: two parity checks Quantum: two non-demolition measurements, accomplished with CNOT circuit
The three-bit repetition code
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A group theory for codes: stabilizers
P C(S) S
P
Groups: 1) Pauli group P: all products of Pauli operators on a set of qubits, e.g., IXZXXYZ…
2) Stabilizer group S: abelian subgroup of P. States protected by code are +1 eigenstates of S
3) Centralizer C(S): operators in P that commute with all operators in S. (a non-‐abelian group).
The theorem: errors can be detected as long as they are not in C(S)\S.
Calderbank-‐Shor [also Goaesman], 1997
This assured that there would be a huge number of codes.
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M(1)=Z1Z2Z3Z7!M(2)=Z1Z2Z4Z6!M(3)=Z1Z3Z4Z5!M(4)=X1X2X3X7!M(5)=X1X2X4X6!M(6)=X1X3X4X5!
Error correction: circuit does non-demolition measurement of operators
|0>
anci
lla
The early favorite: Steane 7-qubit code
Most efficient CSS code that corrects one general quantum error (X, Y, Z) All gates are essentially CNOTs
Distressingly difficult experiment! Lots of qubits, lots of long-distance coupling (regularity is not geometric)
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Analysis of fault tolerance:
TNp
Consider algorithm requiring N qubits and T time steps. Without error correction, the probability of failure for a run of the algorithm is estimated as
Not small unless p<10-15 for runs of interest. Consider a code which will correct one error, so that peff= Cp2. C is a couting factor near 10,000. Now the probability of failure is
TNCp2
Slightly improved for small p, but not good enough. But there are many codes, Including ones that correct x errors. We can choose x with a knowledge of N. So the failure probability becomes
T(N)NC[x(N)] p x(N)+1 = poly(N)C[x(N)] p x(N)+1
So long as C[x] doesn’t grow too fast with x, then x can be chosen such that for some finite p, this expression can always be made <<1.
However:
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However:
C[x]≈ xcx
C[x] ≈ cx;
For many families of codes the counting factor grows incredibly fast with x:
One solution: for special sequences of codes, those produced by concatenation, The scaling is better:
pth ≈ 1/c.
For various codes, this gave pth ≈ 10-4 or 10-5.
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Concatena>on: regular, but produces very non-‐local circuits
-‐gates of coded qubits: transversal construc>on (also geometrically highly nonlocal) Even more to do to get universal computa>on -‐magic states -‐teleported gates
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Other development of 1996-‐7:
X X
X
X
Z Z
Z Z Stabilizer generators XXXX, ZZZZ;
Stars and plaqueaes of interes>ng 2D lahce Hamiltonian model
In Quantum Communica,on, Comput-‐ ing, and Measurement, O. Hirota et al., Eds. (Ple-‐ num, New York, 1997).
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Q Q
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Surface code error correction: qubits (abstract) in fixed 2D square arrangement (“sea of qubits”), only nearest-neighbor coupling are possible
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Surface code
Initialize Z syndrome qubits to
Q Q Q
Implementing the “surface code”: -- in any given patch, independent of the quantum algorithm to be done:
Colorized thanks to Jay Gambetta and John Smolin
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Surface code
CNOT left array
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Surface code
CNOT down array
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Surface code
CNOT right array
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Surface code fabric
CNOT down array
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-- prepare 0+1 state
Surface code fabric
-- measure in 0/1 basis 0/1
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Surface code
Shifted CNOT right array
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Surface code
Shifted CNOT down array
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Surface code
Shifted CNOT left array
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Surface code
Shifted CNOT up array
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+/- +/- +/- +/-
+/- +/-
Surface code fabric
Repeat over and over....
+/-
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S. Bravyi and A. Yu. Kitaev, “Quantum codes on a lattice with boundary,” Quantum Computers and Computing 2, 43-48 (2001). M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes,” Found. Comp. Math. 1, 325 (2001)
With these 13 qubits, one gets a standard code that will correct for one error:
Another view of the 2D Surface Code
4-qubit QND parity measurement:
Red diamond: the same in the conjugate basis
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Calculated fault tolerant threshold: p ≈ 0.7% Crosstalk assumed “very small”, not analyzed Residual errors decrease exponentially with lattice size Gates: CNOT only (can be CPHASE), no one qubit gates If measurements slow: more ancilla qubits needed, no threshold penalty
Observations:
NB: Error threshold for 4-qubit Parity QND measurement is around 2% < p < 12%
Now p > 1 %, according to Wang, Fowler, Hollenberg, Phys. Rev. A 83, 020302(R) (2011)
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To get a qubit: Stop measuring one Plaquette (Freedman and Meyers, 1998)
Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski, “High threshold universal quantum computation on the surface code,” Phys. Rev. A 80, 052312 (2009).
24-qubit structure
8 ZZZZ checks 12 XXXX checks
How to compute with the surface code:
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To get a qubit: Stop measuring one Plaquette (Freedman and Meyers, 1998)
Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski, “High threshold universal quantum computation on the surface code,” Phys. Rev. A 80, 052312 (2009).
Logical gates (X and Z) on the “hole” qubit
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E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,” J. Math. Phys. 43, 4452–4505 (2002).
No ghost defects: 2D random bond Ising, FM/PM transition at 10% error rate With ghosts: 3D random plaquette model, Transition at 1% error rate
Experimentalists can skip the next 6 slides
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Larger hole is also OK. Has greater error- correcting capability
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Convenient to associate a qubit with a pair of holes.
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Holes movable by enlargement and reshrinkage Accomplished by changing paaern of opera>ons/measurements Doesn’t interfere with collec>on of Error-‐correc>on informa>on
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CNOT: braid a hole from one qubit in between the holes embodying the other qubit. Error correction continues throughout gate operation Superior to “tranversal” technique
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Error correction: apply matching-algorithm to data in this space-time volume
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Nov. 2011
Alternative way to do logic gates on coded qubits: “Lattice Surgery” (alternative to “braiding”)
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Five-‐Squares Code
Qubits are at ver>ces. Periodic boundary condi>ons (torus). G=< #↓% > where #↓% are 2-‐qubit Pauli link operators. • Dashed links are ZZ. • Solid links around squares are XY (clockwise say). • Solid links between squares are ZZ. Note that solid links an>commute when they overlap on one qubit.
Degree-‐3 hypergraph (generaliza>on of Kitaev’s honeycomb model)
Suchara, Bravyi, Terhal, arxiv:1012:0425
Subsystem concept leads to many new codes…
Ke Ke
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Subsystem stabilizer codes Non-‐Abelian group G with elements <Gi> where Gi have, say, local support on a lahce. Abelian center of group S. Operators in C(S) (in but not in G) are logical operators for protected qubits. Operators in G are logical operators for gauge qubits (extra unused qubits)
P C(S) G
P
S
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• Original insights still being played out
• Maybe a good evolutionary path to quantum computer hardware
Conclusion: quantum error correction in your future
Concept (IBM) of surface code fabric with Superconducting qubits and coupling resonators
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39
Fin