Download Shor state

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Quantum Error Correction:
Optimizing Implementations of the [7,1,3] Code
Yaakov S. Weinstein
MITRE Quantum Information Science Group
April 23, 2014
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What Can a Quantum Computer Do?
Solve mathematical problems intractable on classical computers!
Break RSA encryption codes!
Classical Cryptography based
on the RSA code is secure
provided an eavesdropper
cannot factor large numbers.
Shor’s algorithm allows a
quantum computer to factor large
numbers in a short time!
Typical Quantum Mechanic
with his quantum computer
Classical Channel
Alice
RSA encrypted line
Bob
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What Else Can a Quantum Computer Do?
Faster database searches (i.e. pattern matching)!

Ann’s
Bob’s
Charlie’s
CLASSICAL ORACLE
0 or 1
0 or 1
0 or 1
Nth person
Classically ~ N/2 lookups
Ann’s
Bob’s
Charlie’s

QUANTUM ORACLE
1
0  1
12

12
2
0
1

0
1

Nth person

Grover’s algorithm for a quantum
computer ~ N lookups
Other algorithms have been discovered that allow a quantum computer
to perform better than a classical computer.
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How Does a Quantum Computer Work?
Bits vs. Qubits
Bit stored in capacitor
“0 or 1”
Qubit stored in spin ½ particle
phase
-
+
• Two possible states:
charged or unchaged
• Millions of electrons
determine charge state
𝟏
(|𝟎
𝟐
𝟏
(|𝟎
𝟐
|𝟎
-|𝟏 )
𝟏
(|𝟎
𝟐
+i|𝟏
)
-i|𝟏 )
𝟏
(|𝟎
𝟐
+|𝟏 )
|𝟏
• Qubit can be in a
continuum of states
• Most states are
superpositions of 0 and 1
Measuring a qubit ‘collapses’ the state to “0 or 1”
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Superposition
One qubit can be in 2 states at the same time.
Two qubits can be in 4 states at the same time.
n qubits can be in 2n states at the same time!
So…
On a ‘classical’ computer one can apply an algorithm to one input at a
time
On a quantum computer one can put the input into a superposition of a
huge number of states and then apply the algorithm to all inputs
simultaneously!
But…
The measurement outcome is quantum probabilistic. So even though
the algorithm is applied to many states you can only measure one
answer and which one you get is probabilistic!
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Entanglement
Entanglement is even
more strange!
Measur
e

A
B
A
B
A
½ the time
B
All of the time
A
½ the time
B
All of the time
•Superdense Coding - send two classical bits of information via one
qubit
•Quantum Cryptography - guaranteed secret by the laws of physics
•Quantum Teleportation - send quantum information without sending a
physical qubit
•Algorithms - Shor’s factoring algorithm, Grover’s search algorithm,
Others
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Quantum Circuit Paradigm
The gate paradigm is similar to classical computer
architecture. Distinct gates are sequentially applied to
physically recognizable qubits.
Unitary operators:



Two-qubit gates
1
2
3
qubits
One-qubit gates
time
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So Why Don’t We Have a Quantum Computer?
Decoherence – the interaction between a quantum systems
and its
environment causing perceived non-unitary
evolution when
looking at the system dynamics alone.
Decoherence can cause conventional (bit flips) and quantum (dephasing)
errors and tends to destroy superpositions…
Error Correcting – monitor system via syndrome measurements for
feedback control
Many known codes [7,1,3], [5,1,3], surface codes, etc.
Error Avoidance –
Bang-bang control – average out environment effects via open
loop control techniques
Decoherence Free Subspace (DFS)– encode information into
system subspace
Noiseless Subsystem (NS) – encode information into a system
degree of freedom
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𝛼|0 + 𝛽|1
|0
|0
|0
|0
|0
|0
ENCODE
Quantum Error Correction
0𝐿
= |0000000 + |0001111 + |0110011
+ |0111100 + |1010101 + |1011010
+1 |1100110 + |1101001
𝐿
Logical
Basis
States
= |1111111 + |1110000 + |1001100
+ |1000011 + |0101010 + |0100101
0011001code-words,
+ |0010110
Distance+3 |between
one error can be corrected
An [n,k,d] QEC code will encode k qubits of quantum information
in n physical qubits with distance d between code-words
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Quantum Fault Tolerance (QFT)
 Allows for successful computation despite errors in basic components
 This can be done by concatenating quantum error correcting codes
QEC
Level 1
QEC
Level 1
QEC
QEC
QEC
QEC
Level 1
QEC
QEC
QEC
Level 1
QEC
QEC
QEC
Level 1
QEC
QEC
QEC
Level 1
QEC
QEC
QEC
Level 1
QEC
QEC
Level 1
QEC
QEC Level 3
Level
Level
Level2
Level
QEC
1
1
1
QEC
Level 1
QEC
Level 1
QEC
Level
Level
Level2
Level
QEC
1
1
1
Level 1
QEC
Level 1
QEC
QEC
Level 1
QEC
Level
Level
Level2
Level
QEC
1
1
1
QEC
Level 1
QEC
Level 1
QEC
Level
Level
Level2
Level
QEC
1
1
1
QEC
Level
1
QEC
Level 1
QEC
Level
Level
Level2
Level
QEC
1
1
1
QEC
Level 1
QEC
Level 1
Level
Level
Level2
Level
QEC
1
1
1
QEC
QEC
Level 1
QEC
Level 1
QEC
Level 1
QEC
Level 1
QEC
Level 1
QEC
Level 1
QEC
Level 1
Then, if you keep your error probability on basic components below
a calculated threshold, you can successfully compute despite errors
QEC
Level 1
QEC
Level 1
QEC
Level 1
QEC
QEC
QEC
Level
Level
Level2
Level
QEC
1
1
1
Single
Useful rules for making
your computation fault tolerant:
qubit
Single qubit encoded
1. Keep your information in the error correcting codeinto
manifold
7 physical qubits
QEC
QEC
QEC
QEC
QEC
QEC
QEC
• Or else errors
will
occur
when
you’re
outside
of
the
correcting
Level 1
Level 1 QEC
Level 1
Level
1
Level2
1
Level 1
Levelerror
1
Level
will protect
against
all space
one qubit errors
2. Do not reuse qubits
Now have 49 qubits
and can protect
3. Do not allow a qubit to interact with more than one other qubit
against two singlequbit errors
4. Repeat to ensure no errors have occurred
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Violations of QFT Rules:
Syndrome Measurements
How do we
construct these?
(no Hadamards)
Shor State
(no Hadamards)
Shor State
(no Hadamards)
Shor State
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Violations of QFT Rules:
Encoding
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Error Model
Non-equiprobable Pauli error environment on single qubit:
𝜌𝑓 = 1 − 𝑝𝑥 − 𝑝𝑦 − 𝑝𝑧 𝜌𝑖 + 𝑝𝑥𝜎𝑥𝜌𝑖𝜎𝑥 + 𝑝𝑦𝜎𝑦𝜌𝑖𝜎𝑦 + 𝑝𝑧 𝜎𝑧𝜌𝑖𝜎𝑧
𝑝0
We assume that any qubit taking part in a quantum gate (including
initialization and measurement) undergoes no errors with probability p0,
an error sx with probability px, sy with probability py, sz with probability pz.
Where sj are the Pauli spin matrices and s0 is the identity matrix.
CNOT gate
errors: p0, px, py, pz
0,𝑥,𝑦,𝑧
errors: p0, px, py, pz
no errors
𝑝𝑎 𝑝𝑏 𝜎𝑎𝑗𝜎𝑏𝑘𝐶𝑗 𝑁𝑂𝑇𝑘 𝜌𝑖 𝐶𝑗 𝑁𝑂𝑇𝑘𝜎𝑎𝑗𝜎𝑏𝑘
𝜌𝑓 =
𝑎,𝑏
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Shor State Construction
Parity Check
GHZ ConstructionVerification
|0
|0
|0
|0
A Shor state appropriate for the
syndrome measurement of a [[7,1,3]] code
is done by building a highly-entangled
GHZ state. This state is verified to ensure
multiple errors have not taken place
during construction. Hadamard gates
complete the Shor state construction.
Y.S. Weinstein and S. D. Buchbinder,
PRA 86, 052336 (2012);
How many verifications steps should we use? Or is it even worth using
Shor states in the first place – maybe stick with one-qubit ancillae?
Logical Zero:
1 Qubit
No verifications
1 verification
2 verifications
7-Qubit fidelity
1-25px28py-21pz
1-85px-37py-12pz
1-55px-19py12pz
1-55px-19py12pz
1-Qubit fidelity
1-9px-9py
1-25px-11py
1-19px-7py
1-19px-7py
After perfect QEC
1-12px2
1-92px2-74pxpy-14py2
1
1
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Encoding a Qubit in a [7,1,3] Code
(no Hadamards)
Shor State
(no Hadamards)
Shor State
(no Hadamards)
Shor State
But how do we encode our information without error?
Two methods:
1) Apply a gate sequence that will properly encode the information we
wish to preserve (and hope that the errors are not too bad) – not fault
tolerant – errors can spread
2) Apply fault tolerant error correction to seven qubits all in the state zero
state, projecting it into the code space
S.D. Buchbinder, C.L. Huang, and Y.S. Weinstein, QIP 12, 699 (2013);
Y. S. Weinstein, PRA 84, 012323 (2011).
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Gate Fidelities
0
0
0
Perfect
DECODE
0
Noisy
GATE
0
Perfect
ENCODE

0
7-Qubit Fidelity
(fidelity of
process)
OR
Perfect
Error Correction
1-Qubit Fidelity
(fidelity of protected information)
After Perfect QEC Fidelity
(correctability of errors)
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Clifford Gates
For codes such at the [[7,1,3]] code Clifford gates, C, such as the NOT-gate,
Phase gate, Hadamard gate, and CNOT gate, can be implemented bit-wise:
C†
C†
Clifford Gate
7-qubit fidelity
After Perfect QEC
C†
Hadamard
1-7px-7py-7pz
1
NOT
1-3px-3py-3pz
1
Phase
1-7px-7py-7pz
1
C†
C†
C†
C†
One logical qubit fidelities are of the same order as the 7-qubit fidelities but
have additional terms dependent on the initial encoded state.
We can also determine single logical qubit cmatrices which allows us to calculate (state
independent) gate fidelities.
Y. S. Weinstein and S. D. Buchbinder,
J. Mod. Opt., 61, 49 (2014)
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Y.S. Weinstein, PRA
87, 032320 (2013)
Non-Clifford T-Gate
T-Gate
Fault
Tolerant
Logical Zero
Gate Sequence
Logical Zero
7-Qubit Fidelity
1-7px-7py26pz
1-10px-13py-35pz
ZPX
c-Matrix Fidelity
1-3px-5py14pz
1-6px-11py-23pz
1
1
Noisy Encode
Logical Zero
ZPX
ZPX
ZPX
After Perfect
QEC
ZPX
T L
ZPX
X
X
MEASURE
X2
MEASURE
Noisy 7-Qubit
Shor State

Perfect Encoding
of Arbitrary State
ZPX
X
X
X
X
X
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How Often Quantum Error Correction?
|0
|0
|0
DECODE
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Recovery
|0
GATE
|0
ENCODE
𝛼|0 + 𝛽|1
Syndrome
Measurement
Apply N times
 Since all gates are imperfect the application of the syndrome measurements
will also introduce errors
 The application of syndrome measurements is very expensive in terms of
time and number of qubits
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Quantum Error Correction During Many Gates
|0
|0
|0
p
p
p
p
p
p
p
p
p
p
p
p
p
p
GATE
|0
p
GATE
|0
p
GATE
|0
ENCODE
𝛼|0 + 𝛽|1
…
p
…
p
p
p
P(one error):
7p-O(p2)
14p-O(p2)
7np-O(p2)
P(two errors):
21p2-O(p3)
84p2-O(p3)
21[n+2
p
𝑛
2
] p2-O(p3)
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Quantum Error Correction During 50 Gates
How often should we apply QEC such that the output state of our
computation is as accurate as possible?
Better or worse to apply QEC after every gate?
Infidelity
Fractional change of
Infidelity:
𝐷 𝐼50 , 𝐼𝑞 =
Y. S. Weinstein, PRA 89, 020301(R) (2014);
Y. S. Weinstein, PRA 88, 012325 (2013);
Y. S. Weinstein, PRA 87, 032320 (2013).
𝐼50 − 𝐼𝑞
𝐼50
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Infidelity Evolution During 50 Gates
Infidelity (to first order)
vs. gate # is fit by line
Infidelity decreases
after T-gate
X – 50
O – 20
□ – 10
◊–2
●–1
▲– 0
Y. S. Weinstein, working on it...
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Conclusions & Ongoing Work
Quantum computers offer revolutionary improvement in computing power due to
the ability of qubits to be in superposition states and to become entangled. Powerful
quantum algorithms already known include Shor’s algorithm to factor large numbers
and Grover’s fast database searches
 Careful explorations of the basic elements involved in implementing fault-tolerant
quantum computation. I have demonstrated means of increasing the accuracy of
these basic elements
 Analysis of how often QEC should be applied during a sequence of gates
Ongoing work – continued exploration of QEC during multiple gates and
especially the effect of T-gates
Ongoing work – exploration of different types of syndrome measurements
Future work – other codes including other concatenated codes and surface codes