MULT90063 Introduction to Quantum Computing
Lecture 17
Simple classical error correction codes, Quantum error correction codes, stabilizer formalism, 5-qubit code, 7-qubit Steane code
Lecture 18
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The more advanced quantum error correction codes, Fault Tolerance, QEC threshold, surface code.
Quantum error correction
MULT90063 Introduction to Quantum Computing
Fault Tolerance and Topological Error Correction Lecture 18
MULT90063 Introduction to Quantum Computing
This lecture we will introduce more advanced error correction for quantum computers:
– Reviewsomeoftheconceptsfromlastlecture – FaultTolerance
– Concatenatingquantumerrorcorrectioncodes – The“threshold”
– Topologicalquantumerrorcorrection: , Chapter 11
Kaye, Chapter 10
Nielsen and Chuang, Chapter 10
MULT90063 Introduction to Quantum Computing
Quantum computing hierarchy
algorithms regime: >>103 qubits
QEC architecture regime: 1000’s qubits
quantum control
& QEC regime: 10’s qubits
Physics regime:
1-2 qubits
E = 2.5 MV/m
E = 5.0 MV/m
computer/systems physics quantum information
MULT90063 Introduction to Quantum Computing
Three qubit phase-flip code
Encode one qubit as three (redundant information)
↵|0i + |1i ! ↵| + ++i + | i
A rogue error occurs (phase flip of first qubit):
↵|0i + |1i ! ↵| + +i + |+ i
Measure correlations (“stabilizers”) between qubits:
X1X2 = 1 X2X3 = +1
Correct errors which have occurred (in this case, phase flip the first qubit):
↵| + +i + |+ i ! ↵| + ++i + | i
MULT90063 Introduction to Quantum Computing
Correcting Both Bit and Phase Errors
IXZZX XIXZZ ZXIXZ ZZXIX
Five qubit code is the smallest QEC which corrects both bit and phase flips.
Seven qubit “Steane” code is a CSS code which allows encoded operations to be easily applied.
XXXXIII XXIIXXI XIXIXIX
ZZ ZZII I ZZ II ZZ I ZI ZI ZI Z
MULT90063 Introduction to Quantum Computing
Logical Gates
Transversal gates:
Hadamard on a single logical qubit
This gate can be operated while leaving the logical qubit encoded, protected by the QEC code.
MULT90063 Introduction to Quantum Computing
Logical CNOT
Encoded Qubit 1
Can also implement CZ, Swap transversally
Encoded Qubit 2
Danger! CNOTs can propagate errors. We need to make sure this happens in a controlled way.
MULT90063 Introduction to Quantum Computing
Fault Tolerance
Strategy: Take the original circuit and replace it with the logical version. In doing so we need to control the spread of errors. Doing this in a way which controls the spread of errors is known as fault tolerance:
Fault tolerant: a single error in any of the QEC procedures causes at most one error in the block of encoded qubits (which can be corrected)
A single error (on a physical qubit) should not propagate to two errors on the same logical qubit, otherwise we would not be able to correct that qubit.
MULT90063 Introduction to Quantum Computing
Transversal CNOT is Fault Tolerant
Encoded Qubit 1
An error here With probability p
Encoded Qubit 2
Propagates to an error here, but is still correctable. Neither logical qubit has more than one error.
Probability of two (uncorrectable) errors in this block is still proportional to p2.
MULT90063 Introduction to Quantum Computing
NOT Fault Tolerant
Consider the following CNOT gate for the 3-qubit bit flip code (000/111)
Encoded Qubit 1
Although this circuit is “correct” (the operation it performs – assuming no error is an encoded CNOT)…
.. a physical single error can cause the second encoded qubit to be uncorrectable. It is not fault tolerant!
Encoded Qubit 2
Care needed! Every operation (including measurement of syndromes) can have errors. Not only do X errors propagate, so do Z errors.
MULT90063 Introduction to Quantum Computing
Transversal gates are Fault Tolerant
Logical X Logical Z Logical S Logical H
Logical CNOT
Not the only way to achieve fault tolerance, but a very useful one!
MULT90063 Introduction to Quantum Computing
Larger distance codes
We have seen some simple error correction codes which correct one error (distance 3 codes). How can we construct quantum error correction codes which correct more than one error?
|0Li ! |00000i |1Li ! |11111i
Distance 5 bit flip code
More errors needed before uncorrectable, leading to a logical error. More physical qubits give more locations for potential errors.
MULT90063 Introduction to Quantum Computing
Concatenated codes
Systematic way to increase the distance of a code. Feed the code back into itself:
|0L2i = p8(|0L0L0L0L0L0L0Li + |1L0L1L0L1L0L1Li + |0L1L1L0L0L1L1Li + |1L1L0L0L1L1L0Li
+ |0L0L0L1L1L1L1Li + |1L0L1L1L0L1L0Li + |0L1L1L1L1L0L0Li + |1L1L0L1L0L0L1Li |1L2i = p8(|1L1L1L1L1L1L1Li + |0L1L0L1L0L1L0Li + |1L0L0L1L1L0L0Li + |0L0L1L1L0L0L1Li
+ |1L1L1L0L0L0Li + |0L1L0L0L1L0L1Li + |1L0L0L0L0L1L1Li + |0L0L1L0L1L1L0Li)
|0Li = p8(|0000000i + |1010101i
+ |0110011i + |1100110i + |0001111i + |1011010i + |0111100i + |1101001i)
This method is known as “concatenation” of error correction codes.
|1Li = p8(|1111111i + |0101010i
+ |1001100i + |0011001i + |111000i + |0100101i + |1000011i + |0010110i)
MULT90063 Introduction to Quantum Computing
Error after different levels of encoding
Logical error rate achieved
pfail = pth(p/pth )2k pth =10-5,p=10-6
k=1: pfail = 10-7
k=2: pfail = 10-9
k=3: pfail = 10-13
MULT90063 Introduction to Quantum Computing
Encoded CNOT error threshold analysis
Error Correction Threshold
Level 0 10-7
ptrans = pgate/10 pmem = pgate/10 Ttrans = Tgate/10 Tmeas = 100 Tgate
(lower bounds)
Level ¥ threshold: 1.5 x 10-6 Level 1 threshold: 7.3 x 10-7
Physical gate error
Effective error
MULT90063 Introduction to Quantum Computing
From physical qubits to logical qubits
Hierarchy of qubits, quantum error correction (QEC) and concatenation
1. Individual physical qubit with external control and measurement
2. Coupling physical qubits to perform logic gates
3. Encoded logical qubit and QEC including classical feed-forward processing
4. Coupling encoded qubits to perform logic gates protected by QEC
5.Recursiveconcatenation of encoding and QEC for fault-tolerant operation
6. Fault-tolerant implementation of quantum circuits over encoded logical qubits
… ……
… … … … … …
… … … … …
Transversal CNOT
æ p ö2k pfail »pthçp ÷
Concatenation: to gain protection p << pth
For a given physical implementation:
How is this achieved? What is the threshold pth?
MULT90063 Introduction to Quantum Computing
The Surface Code
l A topological code suited to solid state
(Kitaev 1997, Raussendorff 2007)
l A remarkably high threshold of ~1%
(Wang et al, 2011)
u Parallel and synchronous control required
MULT90063 Introduction to Quantum Computing
Errors on the surface code
X -1 Z -1 -1
MULT90063 Introduction to Quantum Computing
Chains of Errors
• Errors form chains, can only see syndrome changes (-1) at the ends.
• Minimum weight matching determines the most likely errors.
• Chains greater than half way across the surface can cause failure.
MULT90063 Introduction to Quantum Computing
Logical Operators on the surface code
A logical X operation is a chain of X operations, left to right
A logical Z operation is a chain of Z operations, top to bottom
Logical operations anti-commute (as they should)
MULT90063 Introduction to Quantum Computing
Distance of the Surface Code
4 Z errors
6 X-errors
• Distance of the code is equal to the length of a side.
• Scale up by simply making larger patch of surface code (concatenation not required)
• Topologically defined, so easy to map onto physical architectures
MULT90063 Introduction to Quantum Computing
The Threshold
DS Wang et al
MULT90063 Introduction to Quantum Computing
Mulitple qubits
Code qubits
Logical Ops
Moving qubits Defects are artificial
boundaries
MULT90063 Introduction to Quantum Computing
Requirements for an Error Corrected Shor
Bits in factored number 2000
Number of Logical qubits 4000 required
Number of qubits in surface ~20 million qubits code
Time for one measurement 100 ns
Total time required 26 hours
Research topic: Bring these requirements down!
Experimental proposal
RESEARCH ARTICLE
High-level:
Qubit: uniform nuclear spin
Addressing: electron load (not gate defined wavefn)
Gates: electron load and global ESR/NMR control
Operation: parallel, 60 MHz loading pulses, robust to local variations
Fig. 1. Physical layout of the donor-based surface code quantum computer. (A) The system comprises three layers. The 2D donor q C. Hill et al, Science Advances 2015
Established ESR/NMR sp
resides in the middle layer. A mutually perpendicular (crisscross) pattern of control gates (initially chosen to be 5 nm in width and 30 nm in pit upper and lower planes form a regular grid of (3D) cells. In the upper plane, the control lines alternate as source (S) and gate A (GA), and in th plane, the control gates alternate as drain (D) and gate B (GB). (B) In the middle plane directly below each intersection of S and D lines is a STM f Si:P monolayer quantum dot, which forms the island of a vertically defined SET facilitating electron loading/unloading and readout. (C) A singl
Criss-cross gate array ®
on the long-lived zero-leakage/zero-loss spin states of the spin-1⁄2 P nucleus ( P
upplementary Materials and fig. S1). By virtue of the shared-control lines, this process can be carried out in
STM-based fabrication tolerances (7, 34). When an electron is loaded to qubits in the memory configuration are sufficiently off-reson 3D STM fabrication of array (McKibbin Nanotechnology 2013)
ory state, the qubit states are
a donor, the hyperfine interaction is immediately switched on, and the remain spectators to the process. Single-qubit operations o Hamiltonian of the qubit in this activated configuration becomes and/or electron spins are thus performed via the global appli
the following Hamiltonian (1)
Initial proposal: CNOT dipole coupling (slow) ® developing faster gates (MHz regime)
Hact 1⁄4 !gnmnBzZn þ gmBBzZe þ Asnse ð2Þ
where s! X;Y;Z and the hyperfine interaction for P donors in !g m B X cos w t Y sin w t
). A specific qubit is activated/deactivated by applying volta
proximal gates (S, D, G , G , G !, and G !) to create the local bias condition to load/unload an electron onto the donor or to place the syste
For N qubits, # control lin
multiple locations. Activation switches on the hyperfine interaction on single donors, and spin-spin interactions for neighboring activate allowing single- and two-qubit gates to be carried out via global ESR and/or NMR control. Nonactivated qubits are detuned from these con
Hglobal 1⁄4 gmBBMWðXe cosðwMWtÞ þ Ye sinðwMWtÞÞ
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