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.

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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.

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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

state α|0> - β |1>:

α|0> + β|1> → α (|000> + β|111>)3 + α (|000> - β|111>)3

The logical representation of the basis state is now:

|0>L = (|000> + |111>)(|000> + |111>)(|000> + |111>)

|1>L = (|000> - |111>)(|000> - |111>)(|000> - |111>)

As we know that HZH = X, indicates that prior to sending the

information on a Quantum channel and just after receiving the

Quantum information we need to apply the Hadamard gates

Figure 8.

Figure 10 Quantum Circuit of Phase Flip with phase error [15]

The phase changes are amplitude (or bit flip) changes if we

consider the Hadamard basis. The Hadamard basis satisfies the

condition HX = ZH. It means that in Hadamard basis, the phase

flips turn to be bit flips and vice versa. This helps us in

determining the encoding code that can correct the Phase errors

by mapping H|0> → H3|000> and H|1> → H3|111>. Finally

the Bit Flip code and Phase Flip code are concatenated to

correct them concurrently. This results in 9 qubit code that is

discussed in detailed in section C of this paper.

The Encoder into Bit Flip code QC has been demonstrated

on an IBM Quantum Experience (IBM QE). IBM QE is a cloud

based platform to interact with an existent Quantum Computer

housed in IBM research lab. The QC can be visualized below.

Figure 11 Bit Flip Encoder Code QC on IBM Quantum Experience

Composer.

C. Shor’s Error Code

Effective techniques for quantum error correction were first

developed in 1995 by Shor. Up till then many skeptical

physicists regarded quantum computing as totally impractical.

With the development of error correction techniques, “totally

impractical” was replaced with “extremely difficult.” We will

be using Shor’s algorithm for correcting arbitrarily single qubit

errors, and it also works for two qubit errors.

Shor’s nine qubit code was first quantum code to be

discovered that has property that it allows recovery from an

arbitrary error on any one of the carriers. There are more

efficient codes that exist but nine qubit code demonstrate how

the error recovery process works, in easy way.

The error channel may induce a bit flip, a sign/phase flip, or

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both. If we can design a code that corrects all Xi, and Zi errors,

then it corrects all single qubit errors. To construct such a code,

we can try to combine CBF (This quantum code encodes the state

|0⟩ in the state |000⟩ and state |1⟩ in the state |111⟩) and CPF

(corrects phase-flip errors) First encoding a qubit using CPF and

then encoding each resulting qubit using CBF leads to the nine-

qubit code, which is known as Shor’s nine-qubit code.

The encoding of the Shor code can be understood in two

parts: The first uses the phase flip code in:

|ψ⟩ = α |+ + +⟩ + β |- - -⟩

The second part consists of encoding each of the three

original qubits into three more qubits using the bit flip code:

| + ⟩ → |000⟩ + |111⟩ and | - ⟩ → |000 ⟩ - |111 ⟩

The end result corresponds to the nine-qubit encoding,

| ψ⟩ = α | 0⟩ + β | 1⟩

We often write these states as follows

Shor 9-qubit quantum code circuit. The “inner” part corrects

any single-qubit X-error. The “outer” part corrects any single-

qubit Z-error. Since Y =iXZ, single-qubit Y-errors are also

corrected.

Figure 12 Circuit encodes the Shor’s nine qubit code.

The logical operations on the new qubit are:

X = Z1Z2Z3Z4Z5Z6Z7Z8Z9 for an X-gate and

Z = X1X2X3X4X5X6X7X8X9 for a Z-gate.

In summary, the Shor code is the concatenation of the bit flip

and phase flip error correcting codes. One way of making a

quantum circuit more robust to errors is by concatenating

quantum error-correcting codes to form new, larger codes.

Figure 13 Schematic illustrating concatenations of code.

At the first level, we encode every qubit with some code. At

the second level, each qubit forming the code word gets

encoded again, using the same or a different code. If we use the

same n-qubit code, after two levels, each logical qubit is

effectively encoded by ‘n2’ physical qubits. For ‘k’ encoding

levels, a logical qubit is formed by ‘nk’ physical qubits.

Like the bit and phase flip error correcting codes there are

stabilizers for the Shor code. However, this time, there are eight

total stabilizers which can detect and diagnosis errors Table 3

Table 3. The three-qubit phase flip code procedure is listed

for each of the four possible stabilizer measurement outcomes

The simple way to think about the Shor code is to

breakdown the state | ψ⟩ into three different blocks of logical

qubits; so, as an example, |0⟩ would have three blocks

containing (|000⟩ + |111⟩ ). The error correction procedure

begins by applying the first six stabilizers each of the three

blocks to detect bit flip errors. There are two-bit flip stabilizers

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for each block. After correcting any bit flip error, the last two

stabilizers are applied to each block to detect phase flip errors.

D. Stabilizer Codes

Stabilizer code is a class of quantum error correcting codes.

The general formalism applies broadly and efficiently describes

general rules to construct preparation circuits, correction

circuits and fault-tolerant logical gate operations.

Daniel Gottesman, using the Heisenberg representation for

quantum mechanics, introduced the first formalism in his

doctoral thesis. Quantum states are described in terms of

operators instead of the states as we shall see in examples later.

It is important to keep in mind that we must learn nothing

about the actual state of the code within the coding space as we

make a measurement to find out about the error. Superpositions

of the basis states would be disturbed if something was indeed

learnt during measurement. Hence, even if we would correct the

basis states, we could not correct an arbitrary valid codeword.

Hence, for example, if we wish to learn about the error Ea and

Eb, we measure ⟨ψi|Ea

†Eb|ψi⟩ for all possible errors Ea and Eb.

A state |ψ⟩ is defined to be stabilized by some operator, K,

if it is a +1 eigenstate of K , i.e. K |ψ⟩ = |ψ⟩ . For instance, a

single qubit state |1⟩ is stabilized by the operator σz so that σz|1⟩ = |1⟩.

Hence, we can define a multi-qubit states in the same way.

An N-qubit stabilizer state, |ψ⟩N is then defined by the N

generators of an Abelian (all elements commute) subgroup, G,

of the N-qubit Pauli group, G={Ki |Ki|ψ⟩=|ψ⟩, [Ki,Kj]=0,

∀(i,j)}⊂PN, where PN is the N fold tensor product of all

elements of P, the Pauli group contains

±σI,±iσI,±σx,±iσx,±σy,±iσy,±σz,±iσz.

Many multi-qubit states are in fact stabilizer states, such as

two-qubit Bell states and Greenberger-Horne-Zeilinger (GHZ)

states. For example, bell states can be written in any two

linearly independent generators.

Four Orthogonal Bell States

Each of the four Bell states correspond to the four ± eigenstate

combinations of these two operators, where X is σx and Z is σz

The use of the stabilizer formalism to describe quantum error

correction codes is very useful since it allows easy synthesis of

correction circuits and also allows for quick determination of

what logical operations can be applied directly on encoded data.

The link between stabilizer codes and stabilizer states is defined

as a related coding subspace within the larger Hilbert space of

a multi-qubit system.

Gottesman states that “…a quantum code with stabilizer S

will detect all errors E that are either in S or anticommute with

some element of S. In other words, E ∈ S ∪ (G − N(S)). This

code will correct any set of errors {Ei} iff EaEb ∈ S ∪ (G −

N(S)) ∀ Ea,Eb (note that Ea†Eb commutes with M ∈ G iff EaEb

= ±Ea†Eb does).”

Despite these advantages, larger codes generally require

more complicated error correction circuits. As the complexity

of the error correction circuits increase, adapting such codes to

physical computer architectures becomes more difficult.

Example: The 7-qubit Steane Code

This is defined as a [[7,1,3]] quantum code, where there are

7 physical qubits that encode 1 logical qubit with a distance

between 3 basis states and correct 1 error.

Each of K above is stabilized by its respective operations.

The set above defines a 2-dimensional subspace and hence a

single encoded qubit. A final operator fixes the encoded state.

For the Steane Code, it is Z = ZZZZZZZ = Z⊗7, and Z |0⟩ L =

|0⟩ L and Z |1⟩ L = − |1⟩ L. |0⟩ L and |1⟩ L are two valid basis

states in state vector, i.e.

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Quantum circuit for preparing [[7,1,3]] is shown below.

Figure 13 Quantum Circuit of 7-qubit Steane Code

VI. CONCLUSION

The intent of this paper is to introduce readers with the basic

understanding of qubits and the likelihood of errors in Quantum

Information System (QIS). These errors in QIS are addressed

with the appropriate quantum error correction measures. The

various noise errors in QIS are described using the relevant

types and their associated error correction approaches. Further

the paper illustrates the Encoder in Bit Flip code on an IBM QE.

This facilitates the reader a more valuable and practical insights

towards the QEC. We encourage them to explore the essential

QEC and emerging fault tolerance techniques further deep and

practically indulge in QEC using the cloud based platforms

such as IBM QE where one can access a Quantum Computer

virtually.

VII. ACKNOWLEDGMENT

The authors would like to thank Dr. Charles Tappert, Dr.

Ronald Frank, and Prof. Istvan Barabasi for creating the CS837

Quantum Computing course at Pace University. Also we would

like to extend our sincere gratitude to IBM Corporation for

conducting Quantum Workshop at IBM Research Centre York

Town, NY and providing academic support and access to the

IBM Quantum Experience.

VIII. REFERENCES

[1] A.M. Steane. Error Correcting Codes in Quantum Theory.

Phys. Rev. Lett., 77:793, 1996

[2] Aoki, T. et al. Quantum error correction beyond qubits. Nat.

Phys. 5, 541–546 (2009).

[3] A.R. Calderbank, E.M. Rains, P.W. Shor, and N.J.A.

Sloane. Quantum Error Correction via Codes Over GF(4). IEEE

Trans. Inform. Theory, 44:1369, 1998.

[4] Chiaverini, J. et al. Realization of quantum error correction.

Nature 432, 602–605 (2004).

[5] D. Gottesman. Fault-Tolerant Quantum Computation with

Local Gates. J. Mod. Opt., 47:333, 2000.

[6] D. Gottesman. A Class of Quantum Error-Correction Codes

Saturating the Quantum Hamming Bound. Phys. Rev. A.,

54:1131, 1990.

[7] Devitt, Simon J., William J. Munro, and Kae Nemoto.

"Quantum Error Correction for Beginners." Reports on

Progress in Physics 76.7 (2013): 076001.

[8] E. Knill and R. Laflamme. A Theory of Quantum Error-

Correcting Codes. Phys. Rev. Lett., 84:2525, 2000.

[9] E. Knill, R. Laflamme, and G.J. Milburn. A Scheme for

Efficient Quantum Computation with linear optics.

Nature (London), 409:46, 2001.

[10] E. Knill, R. Laflamme, and W.H. Zurek. Resilient

Quantum Computation. Science, 279:342, 1998.

[11] Gottesman, Daniel. "Stabilizer codes and quantum error

correction." arXiv preprint quant-ph/9705052 (1997).

[12] Lidar, D. A., Chuang, I. L., & Whaley, K. B. (1998).

Decoherence-free subspaces for quantum computation.

Physical Review Letters, 81(12), 2594.

[13] Ofek, N., Petrenko, A., Heeres, R., Reinhold, P., Leghtas,

Z., Vlastakis, B., ... & Mirrahimi, M. (2016). Demonstrating

quantum error correction that extends the lifetime of quantum

information. arXiv preprint arXiv:1602.04768.

[14] P.W. Shor. Scheme for reducing decoherence in quantum

computer memory. Phys. Rev. A., 52:R2493, 1995

[15] “Qubit.” Wikipedia, Wikimedia Foundation, 21 Mar. 2019,

en.wikipedia.org/wiki/Qubit.

[16] R. Cleve and D. Gottesman. Efficient Computations of

Encodings for Quantum Error Correction. Phys. Rev. A., 56:76,

1997.

[17] Schindler, P. et al. Experimental Repetitive Quantum Error

Correction. Science 332, 1059–1061 (2011).

IX. APPENDICES

The following Python code introduces the implementation of

the various Quantum Error Correction methods. The

environment used is cloud based Jupiter Hub, running with

Python version 3 Notebook. The library used for simulating the

dynamics of open Quantum systems is QuTip. QuTip

essentially provides the user-friendly and efficient numerical

simulations of Quantum Computing. Although there is reusable

code for different error methods at few instances, we try to

8

present each scenario individually with the relevant method.

A. Bit Flip and Phase Flip

Consider the following code. This is the common

declaration for representing all the classes of errors which

import the Qutip Library, and ‘random’ is python package that

generates pseudo random numbers. This will make all functions

and classes in QuTip available in the rest of the program.

Snippet 1. Common Declarations

A state psi is defined which is arbitrary, with two subspaces

that are linearly independent can be noted in Snippet 2. The

‘psi’ represents the logical qubit to be encoded whereas the

‘qubit1’ and ‘qubit2’ denote the physical qubits.

Snippet 2. Physical and Logical Qubit States

Controlled NOT (CNOT) gates are applied to entangle the

EPR state ‘Ψ’. A swap gate is equivalent to three CNOT gates

applied in the given format, so it is given ‘cnot(N=3)’. The

control and target qubits are specified using the keyword

arguments ‘control’ and ‘target’. Syndrome Measurement

variables A and B are assigned since they are the tensor product

of the states of the Pauli operators. The ‘sigmaz’ represents the

Pauli Z operator that acts on the qubits. The ‘qeye’ generates

the identity operator with two quantum states. (From Snippet 2)

Now the qubits are flipped at random using ‘flip’ and

‘random’ functions and the Syndrome is measured for possible

detection of Bit Flips as shown in Snippet 3. This is constrained

to generate Bit Flips.

Snippet 3. Flipping random Qubit and Syndrome measurement

On the other hand, phase errors are introduced at random in

the exclusive python notebook for detecting Phase Flips. The

following Snippet 4 shows determining the basis states as ‘+’

and ‘-’ to introduce the phase factor.

Snippet 4. Converting basis to ‘+’ ’-’ and Changing Phase of Qubit

for Phase Flip implementation

Both the qubits states are then checked using different

possible combinations of ‘a’ and ‘b’ states for the detection of

Bit Flip and Phase Flip. If there is an error detected, it is

reported as -1 and the relevant correction is applied using the

Pauli X and Z operators for Bit and Phase flips respectively.

The code that is responsible for Error Detection and correction

is shown below in Snippet 5.

Snippet 5. Error Detection

After successful error correction, the logical qubit is decoded

in compliance with the schematic process of QEC Snippet 6.

The CNOT gates are applied back to disentangle the state.

Snippet 6. Error Correction and Decode

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B. Shor’s 9-qubit code

First a state ‘psi’ is defined which is arbitrary, with two

subspaces which are linearly independent as in Snippet 1.Then

single logical qubit is encoded with 9 physical qubits. The full

multipartite state of the system is defined and CNOT gates are

applied for encoding that can be inferred in Snippet 7. Finally

the encoding of the qubits is performed using the Phase Flip

code.

Snippet 7. Encode Logical Qubit and define state

Once the encoding of the 9-qubits is performed, random Bit

Flip and Phase Flip errors are introduced in code Snippet 8.

Snippet 8. Introducing Random Bit and Phase Flip Errors

The Bit Flip error detection and error correction are

performed for each block of 3, as the 9-qubits are divided in to

3 blocks. Block level operations are defined and Syndrome

Information is measured on each block of 3 in Snippet 9.

Snippet 9. Detect and Correct Bit Flip Errors

Similarly, the above process is repeated for the Phase Flip

detection and error correction. It is measured across the 3 blocks

as shown in Snippet 10.

Snippet 10. Detect and Correct Phase Flip Errors

To conclude the Shor’s code, the Bit Flip is decoded using

the CNOT operators. While for the Phase Flip the CNOT and

Hadamard operators are applied for successfully decoding the

states.

Snippet 11. Decoding Bit Flip and Phase Flip on Qubits.