Error Analysis for

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Error Analysis for Encoding a Qubit in an
Oscillator S. Glancy and E. Knill National
Institute of Standards and Technology, Boulder,
Colorado, USA
Review of Gottesman, Kitaev, and Preskill (GKP)
scheme Error correcting scheme with naive
threshold Constraints on input qubit states
Proposals for state preparation References GKP,
quant-ph/0008040 Glancy and Knill,
quant-ph/0510107
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GKP Review
An LOQC with squeezing Encodes a qubit (or
qudit) in quadratures of a light mode
p
x
p
x
p
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GKP Review
Error model - shifts in x-quadrature -
shifts in p-quadrature - products of these form
a complete set, spanning all possible
errors Measurement - For Z measurement
use homodyne to find x - For X measurement use
homodyne to find p - can discriminate qubit
states if
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GKP Review
Code words are stabilized by with logical
operators Clifford group operations can be
performed with beam splitters, displacements,
phase shifts, squeezing, and homodyne
measurement. Escape Clifford group with cubic
phase state
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GKP Review
GKP virtues - qubits made of light - natural,
simple error correction - homodyne
measurement - linear optical logic gates which
always succeed
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GKP Review
Perfect qubit states are unphysical, containing
infinite energy! Approximate qubit states -
Gaussian peaks of width ? contained in a Gaussian
envelope of width 1/k. - for ?k1/4
These states overlap. For ?k1/4, P0?110-6.
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Error Correction
To correct x shifts x2 measurement gives
knowledge of errors u1 and u2. We then shift
by Qubit becomes p-shift errors add.
x-shift is corrected if
. Then the qubits error is traded for the
ancillas. For larger errors this applies XL.
squeeze
errors
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Error Correction
To correct p shifts p2 measurement gives
knowledge of errors v1 and v2. We then shift by
s(p2). Qubit becomes x-shift errors add.
p-shift is corrected if
. Then the qubits error is traded for the
ancillas. For larger errors this applies ZL.
errors
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Error Correction
State Preparation Threshold - Assume only
errors are in preparation of qubits and
ancillas. - We correct x- and p-shifts. -
Assume worst case addition of errors. - If
all errors have magnitude less than
, then all will be corrected. - We ignored
cases when errors may cancel one another.
Fix x if
Fix p if
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Qubit Error Probability
Probability for a state to have an x-shift of u
and a p-shift of v? Basis states x- and
p-shifts of -Express any state using
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Qubit Error Probability
The probability density for shifts u and v from
is - Examples - Probability that
an input state has error below
is - Pno errors 0.8196 for ?k1/4.
P(u,v)
P(u,v)
v
u
v
u
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Qubit Error Probability
Make a qubit with lower error probability by
reducing ? and k. What is the mean number
of photons in such a state? These approximate
GKP states require large numbers of photons.
Perror
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Qubit Error Probability
To correct displacements, states must have
large . The GKP code can correct phase
errors, because displacements span all errors,
but Phase errors rotate phase space
Phase errors of large states will cause
large displacement errors. How can we balance
reducing intrinsic displacement errors with phase
errors?

p
p
Large displacement
Small phase error
x
x
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Qubit Preparation
Preparing qubits requires non-Gaussian
operations. No useful method using photon
counting and post-selection We need non-linear
coupling. GKPs scheme - prepare
a highly squeezed state. - couple light to
mirror through - measure mirrors phase to
determine lights - apply displacement to get
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Qubit Preparation
Pirandola, Mancini, Vitali, and Tombesis
scheme - use cross-Kerr nonlinearity -
with perfect components measurement confusion
probability P0?11 success probability
2 quant-ph/0402202
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Cubic Phase State
Ancilla cubic phase state allows escaping the
Clifford group. For r?8, this gives
perfect cubic phase states. Using realistic
squeezing r1.34 (12 dB) and perfect photon
counter, fidelity will be 20. High fidelity
cubic phase state requires tremendous squeezing
and excellent counting of large numbers of
photons. Ghose and Sanders, talk at SQuInT
2006, to be published
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Conclusions
If all GKP qubits and ancilla have shift errors
smaller than , then shifts can be
corrected. Approximate qubits with small shift
errors must have large . Large
states are more susceptible to phase errors.
GKP and cubic phase states will be very difficult
to prepare.
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Future of GKP
Open theory questions - Other possible
approximate GKP states? - How to balance shift
and phase errors? - Even a bad approximation of
cubic phase state allows extra- Clifford
operations. Can we make these useful? -
Smarter, easier preparation procedures?
Technology needs - Homodyne detection /
tomography kit - Strong, pulsed, clean
squeezing - Strong, low-loss Kerr effect -
many, many, others References GKP,
quant-ph/0008040 Glancy and Knill,
quant-ph/0510107
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Qubit Preparation
Massive qubit schemes - Trapped ion qubit
encoded in harmonic motion using coupling to
electronic states and measurement. Travaglione
and Milburn quant-ph/0205114. - Neutral atoms
in cavities qubit encoded in atoms motion using
coupling to light in cavity and homodyne
measurement of light. Pirandola, Mancini,
Vitali, and Tombesis quant-ph/0503003,
quant-ph/05100053
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Qubit Preparation
Squeezed Cat scheme - Given a source of cat
states - squeeze and displace each
cat - makes perfect GKP states with
infinite squeezing and iterations. Glancy and
Knill unpublished
iterate