Indian Institute of Technology, Kanpur CS682A Quantum Computation Quantum Error Correction Author: Roll Numbers: Pranav Bisht 17111268 Samik Some 17111271 November 15, 2017 Contents 1 Introduction . .1 2 Bit flip code . .2 2.1 Encoding: . .2 2.2 Error Detection: . .3 2.3 Error Recovery: . .4 3 Phase Flip Code . .4 4 Alternate Look At Syndrome Measurements . .5 5 The Shor Code [Sho95] . .6 5.1 Bit Flip Errors . .7 5.2 Phase Flip Errors . .7 5.3 Combined Bit and Phase Flip Errors . .8 6 Classical Linear Codes . .9 6.1 Generator Matrix . .9 6.2 Parity Check Matrix . .9 6.3 Error Correction . 10 7 CSS codes . 11 7.1 Error Detection and Correction . 12 8 The Steane Code [Ste96] . 14 9 Conclusion . 15 1 Abstract This document serves as a project report on Quantum Error Correcting codes (QEC) done as part of the course Quantum Computation. Efficient and reliable communication has posed a challenge to mankind since ages, eventually leading to birth of Information Technology. This field received a great boost after the advent of Internet with high demand for time and space efficient error correcting codes. Now, at the dawn of Quantum computation, which with every day is coming closer to being a feasible reality, researchers have explored error correcting schemes for reliable transfer of qubits. We primarily focus on Quantum codes and study the classical codes briefly to the extent required for the presentation here. We start with simple repetition code and its quantum equivalent, the bit flip and phase flip codes. We show how Shor code combines both of these. We also look at a broader class of QECs called CSS codes and study Steane code in its light. Pranav Bisht Quantum Error Correction Samik Some 1 Introduction In the classical world, error correcting codes have been extensively studied and developed to ensure reliable communication across a noisy channel. Variety of error correcting schemes exist catering to different requirements like Parity codes, Hamming Codes, Reed Solomon codes et cetera. The primary requirements these codes try to achieve are that of time efficiency with respect to encoding and decoding a message, while also trying to save on the number of extra bits used in the transfer of message. However, in this report our primary focus is to explore Error Correcting algorithms in the Quantum world, highlighting the potential challenges they face and how they are cleverly handled. Quantum error correcting codes derive main idea from their classical counterparts. We will briefly discuss simple repetition code and classical linear codes [6], in order to understand QECs. We will be discussing qubit flip codes [2], phase flip codes [3], Shor's code [5], Steane's code [8] and the general framework of Calderbank-Shor-Steane (CSS) codes [7] in detail in this report. Every error correction scheme has 3 parts - Message Encoding, Error Detection, Error Correc- tion. In all these codes, we will try to describe all the 3 parts in one flow. On the way we will state and assume certain mathematical properties but we will prove them at the end of each section as lemmas, for ease of reading. The setting is as follows: Alice wants to send Bob a message. We can assume the message to be composed of bits (qubits in case of QEC). In real world the communication channel (air, wire) is noisy. Physical factors tend to corrupt the bits. Bob receives a corrupted or original message, but he does not know. How can Bob recover the original message? It is definitely not possible if the message was n bits and Alice sends exactly n bits. However, if Alice sends extra bits, Bob may be able to recover the message. How? Consider Alice wants to send a single bit 0 to Bob. Suppose we are assured that only a single bit error can occur during transmission. If Alice sends 000 as message. Then Bob can receive one of the following: 000 // no error 100 // error on 1st bit 010 // error on 2nd bit 001 // error on 3rd bit On receiving the message, Bob simply takes the majority (max of 0,1) in order to get the original message (in this case 0). Similarly, if Alice wished to send 1 she would instead send 111, and Bob could decode the message appropriately. This simple idea is exploited in Repetition codes. The key idea in any kind of error correction scheme is to append “sufficient” number of redundant bits to the message. Researchers are always trying to optimize the extra bits needed. In fact, there is a dedicated field known as Coding Theory which studies these codes in detail providing theoretical bounds alongside. 1 Pranav Bisht Quantum Error Correction Samik Some Let us now think about QECs. Alice wants to send Bob qubits. Noise in the channel corrupts qubits. How does Bob recover the original qubits? A careful thinker will be stumped by these challenges first: • No cloning: The moment we hear repeated copies, it should raise an alarm in the quantum world, because of the no cloning theorem. The laws of quantum mechanics don't allow copying of qubits. So we cannot simply send 3 copies of qubits to send a single qubit reliably. • Continuous errors: In the classical world, the only kind of error is bit flip. But here a continuum of different errors may occur, due to infinite superposition states possible. Deciphering the order of errors seems to require infinite precision. • Measurement destroys the state: In classical error correction, Bob observes the message received and adopts decoding procedure accordingly. But in the case of qubits observing or measuring perturbs the quantum state. Despite these potential challenges, we will see how researchers deftly overcome these in the error correcting codes to follow. Also, in this writing, we assume that no errors are introduced while encoding or decoding which in the real world does happen. Nonetheless, the theory of fault tolerant quantum computation ensures that they are not a significant setback. 2 Bit flip code As of now, assume that there are only two types of qubit errors: bit flip and phase flip, described by unitary matrices X and Z respectively. Throughout this report, we will concern ourselves with reliable transmission of just a single qubit from Alice to Bob. In this section, we will show how to correct single qubit bit flip error and in the next section we will show the same for qubit phase flip error. Later we will discuss why we can correct any kind of quantum error if we are able to correct both bit flip and phase flip errors. Single bit flip error is described by: j0i ! j1i j1i ! j0i This is the same as action of Pauli matrix X. 2.1 Encoding: Suppose Alice wants to send qubit Ψ = a j0i + b j1i. Similar to the repetition code, we will append two qubits j00i to Ψ, giving us Ψ1. Note that we bypassed the first challenge as we did not copy the state Ψ. We merely appended two ancilla bits which is always possible. 2 Pranav Bisht Quantum Error Correction Samik Some Figure 1: Bit Flip Code Encoding [Wik17] Ψ0 ≡ (a j0i + b j1i) j0i j0i ≡ a j000i + b j100i Ψ1 ≡ a j000i + b j111i 2.2 Error Detection: Assuming that only a single qubit flip can occur during transmission, Bob will receive one of these 4 possible states: Φ0 ≡ a j000i + b j111i // No error st Φ1 ≡ a j100i + b j011i // 1 qubit flip nd Φ2 ≡ a j010i + b j101i // 2 qubit flip rd Φ3 ≡ a j001i + b j110i // 3 qubit flip Let us define the following 4 Projection operators: P0 ≡ j000i h000j + j111i h111j // No error st P1 ≡ j100i h100j + j011i h011j // 1 qubit flip nd P2 ≡ j010i h010j + j101i h101j // 2 qubit flip rd P3 ≡ j001i h001j + j110i h110j // 3 qubit flip Define measurement M = 0P0 + 1P1 + 2P2 + 3P3. Lemma 2.1 proves that fP0;P1;P2;P3g indeed form a valid measurement. Observe that MΦi = i 8i = 0 ::: 3, since ( 1 if i = j Pi jΦji = 0 otherwise This detects which qubit was flipped during the transmission. Note that we chose our pro- jections for measurement cleverly so that the state before and after the measurement is same jPiΦii Φi = 2 . In this way the third challenge is taken care of by measuring in superposition. kPiΦik 3 Pranav Bisht Quantum Error Correction Samik Some Figure 2: Bit Flip Code Encoding and Decoding [Wik17] 2.3 Error Recovery: Once the error syndrome is detected, error recovery is very simple: • No error (0)) Do nothing. • 1st qubit flip (1)) Flip 1st qubit again. • 2nd qubit flip (2)) Flip 2nd qubit again. • 3rd qubit flip (3)) Flip 3rd qubit again. The overall encoding, decoding quantum circuit is shown in Figure 2 Lemma 2.1. The four projectors P0;P1;P2;P3 defined in section 2.2 form a valid measurement Proof. It is easy to verify that 3 X Pi = I8×8 i=0 Pi · Pj = 0 ; 8i 6= j Hence, they form a valid projective measurement. 3 Phase Flip Code Now, we will see how to handle only a single qubit phase flip which might creep in during transmission. Phase flip error is equivalent to action of Pauli matrix Z. a j0i + b j1i ! a j0i − b j1i Here Hadamard gate comes to our rescue again. The neat idea is that phase flip error in the j+i ; |−i basis is same as bit flip error: j+i ! |−i |−i ! j+i This simple idea gives the whole algorithm for phase flip errors: 4 Pranav Bisht Quantum Error Correction Samik Some Figure 3: Phase Flip Code Encoding [Wik17] Figure 4: Phase Flip Encoding and Decoding [Wik17] • Append ancilla bits: a j0i + b j1i ! a j000i + b j111i • Apply Hadamard: a j000i + b j111i ! a j+ + +i + b |− − −i • Apply bit flip algorithm to detect and correct a bit flip error in the j+i ; |−i basis.