1 Abstract— Quantum computing is an emerging technology. Similar to classical computing where transmitting data over network or in memory could corrupt bit strings, current quantum computer technology is not immune from data corruption. Different types of quantum computers are built to exhibit quantum properties. In ideal condition, quantum device should show exact realizations of quantum information and data manipulation, but in real world, it doesn’t behave as theoretical model mostly due to environmental noise. Since 1995, many new mythologies, codes and techniques have been developed to mitigate errors in large scale quantum algorithm. This paper is widely associated with the information in Qubits that emphasizes the prominence of the Error Correction for a sustained and reliable Quantum Information System. It navigates through the basic information about the Qubits and advertently considers the various approaches towards Quantum Error Correction. Index Terms— Quantum Computing, Quantum Error correction, Quantum Circuits I. INTRODUCTION UANTUM Error Correction (QEC) has been introduce in 1995 and is concerned with the reliable Quantum Information Processing in the presence of local errors. While the classical computing relies on Parity for Error correction, the Quantum Computing is farfetched and meticulous than that. In order to understand the level of complexity involved we begin with some basic information about the interpretation of Quantum Information System and build on the theory revolving around the QEC. II. QUANTUM COMPUTING BASIC Qubit is the basic unit of quantum computing [15]. In contrast to the classical binary bits of information in ordinary computers, “qubits” consist of quantum particles that have some probability of being in each of two states, 0 or 1, at the same time as shown in Figure 1. A single qubit is represented by basis vectors |0> and |1>. In Quantum Information Theory the states |0> and |1> form an orthonormal basis of the Hilbert Space of a single Qubit. The Hilbert Space in fact includes all linear combinations of basis |0> and |1> [7]. In pure qubit state, a single qubit can be described by a linear combination of |0> and |1>: A. Measurement The quantum state of a qubit is unknown until it is measured. After the measurement, it changes the state and the process is an irreversible operation. The probability of outcome |0> with value 0 is |α| 2 and the probability of outcome |1> with value 1 is |β| 2 . Since there are two outcomes, |α| 2 + |β| 2 is always equal to 1. See Figure 2. Figure 1. Comparison between classical bit and qubit bit. Figure 2. Single Qubit Measurement Results. B. Operation on qubit states Qubits manipulation takes place in quantum logical gates. And these quantum logical gates are building blocks of a quantum circuit in a quantum computer. The most common single qubit gate is NOT-gate. CNOT-gate is one of the most important 2-quibit gates besides Hadamard ‘H’ gate. A Quantum Circuit with Hadamard ‘H’ and CNOT-gates is shown in figure 3. Mohammad Z Chowdhury, Krishna Bathula, Lu Dong, Kaleemunnisa LNU, and Charles C. Tappert Seidenberg School of CSIS Pace University, Pleasantville NY 10570 Quantum Error Correction Basics 1 Q Proceedings of Student-Faculty Research Day Conference, CSIS, Pace University, May 3rd, 2019 1 Thanks to the IBM Faculty Award that made this research possible.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Abstract— Quantum computing is an emerging technology.
Similar to classical computing where transmitting data over
network or in memory could corrupt bit strings, current quantum
computer technology is not immune from data corruption.
Different types of quantum computers are built to exhibit
quantum properties. In ideal condition, quantum device should
show exact realizations of quantum information and data
manipulation, but in real world, it doesn’t behave as theoretical
model mostly due to environmental noise. Since 1995, many new
mythologies, codes and techniques have been developed to mitigate
errors in large scale quantum algorithm. This paper is widely
associated with the information in Qubits that emphasizes the
prominence of the Error Correction for a sustained and reliable
Quantum Information System. It navigates through the basic
information about the Qubits and advertently considers the
various approaches towards Quantum Error Correction.
Index Terms— Quantum Computing, Quantum Error
correction, Quantum Circuits
I. INTRODUCTION
UANTUM Error Correction (QEC) has been introduce in
1995 and is concerned with the reliable Quantum
Information Processing in the presence of local errors.
While the classical computing relies on Parity for Error
correction, the Quantum Computing is farfetched and
meticulous than that. In order to understand the level of
complexity involved we begin with some basic information
about the interpretation of Quantum Information System and
build on the theory revolving around the QEC.
II. QUANTUM COMPUTING BASIC
Qubit is the basic unit of quantum computing [15]. In contrast
to the classical binary bits of information in ordinary
computers, “qubits” consist of quantum particles that have
some probability of being in each of two states, 0 or 1, at the
same time as shown in Figure 1. A single qubit is represented
by basis vectors |0> and |1>. In Quantum Information Theory
the states |0> and |1> form an orthonormal basis of the Hilbert
Space of a single Qubit. The Hilbert Space in fact includes all
linear combinations of basis |0> and |1> [7]. In pure qubit state,
a single qubit can be described by a linear combination of |0>
and |1>:
A. Measurement
The quantum state of a qubit is unknown until it is measured.
After the measurement, it changes the state and the process is
an irreversible operation. The probability of outcome |0> with
value 0 is |α|2 and the probability of outcome |1> with value 1
is |β|2. Since there are two outcomes, |α|2 + |β|2 is always equal
to 1. See Figure 2.
Figure 1. Comparison between classical bit and qubit bit.
Figure 2. Single Qubit Measurement Results.
B. Operation on qubit states
Qubits manipulation takes place in quantum logical gates.
And these quantum logical gates are building blocks of a
quantum circuit in a quantum computer. The most common
single qubit gate is NOT-gate. CNOT-gate is one of the most
important 2-quibit gates besides Hadamard ‘H’ gate. A
Quantum Circuit with Hadamard ‘H’ and CNOT-gates is shown
in figure 3.
Mohammad Z Chowdhury, Krishna Bathula,
Lu Dong, Kaleemunnisa LNU, and Charles C. Tappert
Seidenberg School of CSIS Pace University, Pleasantville NY 10570
Quantum Error Correction Basics1
Q
Proceedings of Student-Faculty Research Day Conference, CSIS, Pace University, May 3rd, 2019
1 Thanks to the IBM Faculty Award that made this research possible.
2
Figure 3. A 3-qubit circuit with CNOT-gate and Hadamard-gate.
III. QUANTUM ERROR CORRECTION
Before Shor’s [14] algorithm in 1995, despite quantum
computing’s demonstrated efficiency over classical computing
on small scale, the coherent quantum states were extremely
weak and many believed that it wouldn’t be realistic to maintain
large, multi-qubit and coherent quantum states for a long
enough time to complete any quantum algorithms.
A. Previous and current research
However, classical error correction cannot meet the demand
because firstly, the classical techniques assume all of the bits in
the computer can be measured and this assumption would
destroy any entanglement between qubits; secondly, a classical
computer only needs to store 0 and 1, but a quantum computer
would need to store phase information in entangled states[11].
In the early stage of research, an emphasis was placed on
developing quantum codes ([1],[6],[8]). Researchers then
continued to introduce more demanding theoretical frameworks
for the structure and operation of QEC ([9],[16]),[3]).
Meanwhile, concepts such as fault-tolerant quantum
computation led to the birth of the threshold theorem for
concatenated QEC ([10],[5]). Different approaches to continue
developing QEC protocols have been made in areas such as
ions, photon, etc. ([4],[17],[2]). While a lot of previous work
cited focused on primarily illustrating the scaling properties of
QEC codes, Ofek, et al [13] tried to use QEC to extend the
lifetime of quantum information.
Passive QEC methods, such as error suppression have also
been developed as well. One of the most error suppression
protocols is the decoherence free subspaces [12].
IV. QUANTUM ERRORS: CAUSE AND EFFECTS
Quantum Computers compute at individual level of atoms.
Based on the classical error correction analogy QEC was
developed. The physical mechanism behind quantum system is
responsible for quantum errors to exist. We analyze the errors
and their types in the context of physical system.
A. Environmental decoherence
In quantum system, environmental decoherence is the
important source of error. The problem with Quantum
Computers is, instability caused by environmental interference
that can upset the state of the qubits. Decoherence occurs when
the qubits lose information due to the environment they are in
over time. Sometimes decoherence is referred to noise. Adding
more qubits in the system would increase the overall
vulnerability to decoherence. This makes big obstacle in
scalability.
B. Reliable Execution of gates
Since there is no technique exists to perform gates
transformation with perfect accuracy, there always will have
small imperfections in transformation. Also, incorrect
knowledge of the system dynamics could yield incorrect result.
Therefore, qubit systems must be characterized before being
used so that errors can be controlled coherently.
V. QUANTUM ERROR CORRECTION REQUIREMENTS
Classical coding theory plays an important role in quantum
error correction, yet certain issues need to be considered when
transferring classical technique to quantum domain. Moreover,
classical methods are only designed to correct large discrete
errors (i.e. bit flips), where quantum error is continuous. Due to
No Cloning theorem which prevents making perfect copies of
unknown quantum state of qubits, classical code-based data-
copying can’t be used in QEC. Another important factor is, we
can’t measure the qubits for its error directly as this would
collapse quantum superposition state. Qubits are prone to both
traditional bit errors |0> ↔ |1> and phase errors |0> ↔ |0>, |1>
↔ -|1>. Therefore, to detect and correct errors without
measuring the qubits stats, error correction protocol which is
redundant encoding must be deployed.
A good strategy for constructing a Quantum Code ([1],[6]) is
to ensure no information of the encoded state is present in any
single Qubit. The resulting code keeps all information out of
every pair of Qubits.
The schematic process of QEC in figure 4 involves encoding
of qubit, detection for noise and finally decoding. Encoding
process takes the qubit representing data along with the clean
ancilla qubits in order to produce a code block that correct the
error. Syndrome Measurement extracts the error information in
the qubit from the ancilla qubits without actually disturbing the
qubit state. The decoding process allows in identifying any
errors that occurred when qubits traverse through noisy
channel.
Figure 4 Schematic Process of QEC.
A. Bit Flip
In classical computing, bit flips occur in transmission due to
the noise in wire or in memory due to heat, low voltage and
other environmental factors as shown in figure 5.
3
Figure 5 Classical computing bit flip.
In quantum system, bit-flip occurs due to environmental
decoherence/noise. In this paper, we will discuss how to correct
single qubit bit-flip using 3-qubit bit-flip code. Keep in mind
that, this code can’t correct both bit and phase flips
simultaneously. We will discuss later how this code can be
extended to construct 9-qubit Shor [14] quantum code.
Three physical qubits can be used to represent one logical
qubit of information to correct any single bit-flip error. Hence,
we deploy two ancilla qubits, with the objective of extracting
Syndrome Information out of the data block and simultaneously
not trying to disturb and distinguish the exact state of the actual
qubit.
The two logical basis states |0>L and |1>L are defined as,
|0>L = |000>, |1>L = |111>
Such that an arbitrary single qubit state |ψ> =α|0> + β|1> is
mapped to,
α|0> + β|1> → α|0>L+ β|1> L
= α|000> + β|111>
= |ψ>L
A quantum circuit required to encode a single logical qubit
using two ancilla qubits and two CNOT gate is shown figure 5.
This circuit prepare the |0>L state for the 3-qubit code where an
arbitrary single qubit state, |ψ> is coupled to two initialized
ancilla qubits via CNOT gates to prepare |ψ>L.
Figure 6 An encoded circuit of single logical qubit[7]
The encoding and correcting circuit is illustrated in figure 7.
Detailed technical explanation is beyond the scope of this
paper.
Figure 7 Quantum circuit to encode and correct a single qubit bit-flip
error [7]
Assuming that after encoding, a single bit-flip occurs on one
of the three qubits or not error at all. Two ancilla qubits are then
coupled to data block which only checks the parity between
qubits. Then ancilla are measured. Using this syndrome
information, error can be corrected by applying classically
controlled gate.
Table 1 Syndrome information and its corresponding error
Syndrome Error
00 No error
01 3rd qubit flipped
10 2nd qubit flipped
11 1st qubit flipped
Table 2 Ancilla measurement for single qubit error with the 3-qubit
code. All possible four states correspond to either no error or a bit
flip on one of the three qubits.
Collapsed State Syndrome Error
α|000> + β|111> 00 No error
α|001> + β|110> 01 3rd qubit flipped
α|010> + β|101> 10 2nd qubit flipped
α|100> + β|011> 11 1st qubit flipped
B. Phase Flip
In addition to Bit errors, qubits also face the challenge of
phase errors. The phase errors are encountered, when a qubit’s
basis state picks a phase factor. For instance, |0> → eiΦ|0>,
where ‘Φ’ is the angle during the spin and measured in radians
which is less than 1. This angle has a value pointing that the
quantum errors are continuous when compared to the classical
counterparts. Now the initial state which is |Ψ> = α|0> + β|1>
transforms into |Ψ> = α|0> + β eiΦ|1> when the phase factor is
established. For this reason, we need to have a systematic error
detection and correction.
4
Figure 8 Illustration of spin and Phase Factor in a Qubit.
In Quantum Mechanics relative phases in a superposition are
highly significant. In a ‘Z’ or phase flip error we can obtain |0>
→ Z|0> = |0> and |1> → Z|1> = -|1>. Practically while
addressing the Quantum Circuits, when the phase flip or Pauli
operator ‘Z’ is applied to the qubit |Ψ> = α|0> + β |1>, results
in the state |Ψ> = α|0> - β |1>. Hence |Ψ> = α|0> - β |1> does
not represent the same thing as |Ψ> = α|0> + β |1>, except when
α=0 and β =0.
In view of the fact that the repetition of 000,111 corrects the
Bit Flip error, the repetition of phase +++, --- corrects the Phase
Flip error. The code words |+++> and |---> are used to represent
the logical states |+>L and |->L respectively. When the ancilla
qubits are patched with the original qubit |Ψ> = α|+> + β |->,
the logical state for encoding is represented as |Ψ> = α|+++> +
β |---> for the phase flip error detection and correction.
The encoder of the phase flip is similar to the bit flip with
only exception that Hadamard gates are applied at the end as
shown in below Figure 9.
Figure 9 An Encoding circuit of logical qubit with Hadamard Gate
[15]
To correct both Bit Flip and Phase Flip simultaneously we
can use both codes together which is a valid encoding of the