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Quantum error correcting codes

The document discusses quantum error correcting codes (QECC) to combat bit-flip and phase-flip errors in quantum communications. It covers various QECC schemes, including the three-qubit, five-qubit, seven-qubit, and Shor's nine-qubit codes, explaining their encoding, error detection, correction, and decoding processes. A focus is placed on how qubits can be manipulated and preserved through noisy channels while following principles like the no-cloning theorem.

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QUANTUM ERROR CORRECTING CODES BHASKAR REDDY GURRAM(1774985)
CONTENT: 1. Introduction. 2. three-Qubit bit-flip code and Phase-flip code. 3. Shor’s Nine-qubit code. 4. Seven-qubit QECC. 5. Five-qubit QECC.
INTRODUCTION ● Interaction between a quantum system with environment cause undesirable changes in the state of the quantum system. ● In the case qubits, they appears bit-flip and phase flip errors. ● To reduce such errors,we must build in some sort of error correcting mechanism in algorithm ● Before we introduce quantum error correcting code,we have a brief look at the simplest version of error correcting code in classical bits.
Classical ● Suppose we transmit a series of 0’s and 1’s through a noisy classical channel. ● Each bit assumed to flip independently with a probability P. ● Thus a bit 0 sent through a channel will be received as 0 with probability 1- P,And as 1 with probability P. ● To reduce the channel errors,we may invoke the majority vote. ● Namely,we encode logical 0 by 000 & 1 by 111
Probabilities of receiving: ● When 000 sent through this channel,it will be received as 000 with probability(1-P)^3, ● As 100,010,001---probability is 3P(1-P)^2 ● As 011,101,110---probability is 3P^2(1-P). ● And finally as 111 with probability P^3 Note that the summation of all the probabilities is 1 as it should be.
Bit-flip QECC ● Suppose Alice wants to send a qubit or a series of qubit to Bob through a noisy quantum channel. ● Let be the state she wants to send. ● If she is transmit a series of qubits,she sends them one by one following arguments applies to each of the qubits. ● Let P be the probability with which a qubit is flipped .we assume there are no errors in the channel. ● In other words, the operator X is applied to the qubit with probability P, and consequently the state is mapped to
Encoding ● To reduce error probability,we want to mimic somehow the classical counterpart without using a clone machine.. Let us recall that action of a CNOT gate is ● And therefore it duplicates the control bit whether initial target bit is set we use the fact to triplicate the basic vector as ● Denotes the encoded state (logical qubit),while each constituent qubit is called the physical qubit.
Encoding cont.. ● The code and call each member of a C a codeword. ● Important thing is the state is not triplicated but only the basic vectors are triplicated. ● This redundancy makes it possible to detect the errors in state and correct them in further steps.
Fig 1: Quantum circuit represents the (a) Encode (b) Detect Bit-flip syndrome (c) Make correction to relevant qubit (d) Decode
Description of circuit: ● The gate stands for bit-flip noise. ● Circuit (a) only belongs to Alice and remaining (b),(c),(d) belongs to Bob. ● (a) whose inputs are the state and two ancillary qubits in this state we taken as ● When the target bit is ,the ancillary qubits are left unchanged and output is ● While the input is the output is ● By superposition principle ,the input result in output as promised.
Transmission: ● Now the state is sent through a quantum channel which introduce a bit- flip error with a rate P for each qubit independently. ● We assume that P sufficiently small so that not many errors occur during qubit transmission. ● The received state depends on in which physical qubits the bit-flip occurred. ● Table lists the possible received states and probabilities with which these states are received.
Error syndrome detection and correction: ● Now Bob has to extract from the received state which error occurred during the qubit transmission. ● For this purpose,Bob prepares two ancillary qubits in state as depicted in figure. ● Let be a basic vector Bob received and let A(B) be the output state of first ancillary qubit. ● it seen from the previous table.
Syndrome finding: ● From the figure 1 we got this equation ● Let be the received logical qubit, Both of ancillary qubits are flipped for both The set of two bits called the (error)syndrome.and it tell bob in which physical qubit the error occurred during transmission.
It is important to note that (i) the syndrome is independent of a&b. (ii) the received state is left unchanged ● We have detected an error without measuring the received state. ● This feature is common to all QECC. ● Bob applies correcting procedure to the received state according to the error syndrome he has applied. ● Ignoring multiple error states with small probabilities,we immediately find the following action must be taken
Correction: ● Suppose syndrome is 01,the state bob received is likely to be ● So when we applying the on the received state, If bob receives the the state unfortunately he will obtain, Infact , for any error syndrome, bob obtains either or Latter case occurred when more than 1 bit flipped.
Decoding ● Now bob has corrected an error. What is left for him is to decode the encoded state. ● This is nothing but the inverse transformation of the encoding. ● Suppose bob has received a state with more than one qubit flipped: we will obtain
Miracle of entanglement ● We present some redundant qubits which somehow “triplicate” the original qubit state to be sent without violating the no-cloning theorem. ● Then encoded qubits are sent through a noisy channel,which causes a bit-flip in at most one of the qubits. ● The received state, which may be subject to an error,is then entangled with ancillary qubits which detect what kind error occurred during the state transmission .this result in entangled state ● Principle measure the syndrome and single out an error will be used repeatedly in the rest of this chapter.
Continuous rotation: ● We have considered noise X so far, suppose noise in the channel is characterized by a continuous parameter as,and which maps the state as, For example bob receives In the continuous rotation The output of the error syndrome detection circuit,before the syndrome measurement is made, is an entanglement state,
Phase-flip QECC ● Let us consider a phase -flip channel. ● Phase flip occurs with the probability P for each qubit independently when it is sent through a channel. ● We consider how to correct phase flip error mimicking the prescription given from a bit flip-error. ● Phase-flip error in the basis is nothing but Bit-flip in the basis ● The observation suggests that we encode to Now is sent through a noisy channel with possible phase-flip errors.we assume at most a single qubit subject to error.
Block diagram for phase-flip QECC ● By applying the hadamard transform. ● So that a phase-flip error is recognized as bit-flip error and we can employ the same error syndrome detection circuit as well as error correction circuit as those used for bit flip error QECC. ● Collecting these results,the quantum circuit for phase flip QECC is constructed.
Shor’s Nine-qubit code Let us consider a more general noisy channel in which all possible single-qubit error occur. Namely the following errors are now active in the channel. Shor’s nine-qubit QECC which correct the above three types noise is constructed by concatenating the phase-flip QECC into the bit-flip QECC.
● Encoding: we encode ● Bit-flip syndrome detection and correction. ● phase -flip syndrome detection and correction. ● Finally decoding. Fig 2 :Quantum circuit to detect bit- flip and phase-flip syndrome and correct the errors.
Seven-Qubit QECC ● Steane & calderbank & shor proposed a seven-qubit QECC scheme inspired by wisdom of classical error correcting code. ● We index the seven qubits as i=0,1,2,3,4,5,6. ● A possible error is the bit-flip error X and phase-flip error Z independently on one of the seven qubits. ● They act as Y=XZ if they happen to operate on a common qubit.
Encoding and error detection: ● Error detecting circuit of the seven-qubit QECC.the unitary gates controlled by the ancillary qubits correspond to as denoted at the top of the circuit. ● the read out of the six ancillary qubits fixes the set of syndromes.
Decoding: ● Once the syndrome is detected and an appropriate correction is made,we have to decode the logical qubit so as to extract input state ● This is done by applying gates one by one in the reversed order and initial state will be reproduced in the 0th qubit.
Five-qubit QECC ● Divincenzo and shor further reduced the number of qubits required for QECC. ● Suppose n qubits are used to implement one logical qubit with the basis states ● There are 3n operators corresponding 3n single-qubit errors. ● Each operator maps the basic to one of the When n=5 the optimal number to correct all types of single qubit errors.
Encoding and error detection and decoding: ● Encoding: our exposition goes almost parallel to the previous seven-qubit encoding. ● Error syndrome detection:the error detection circuit is constructed strategy employed in the seven-qubit QECC. ● Decoding: once the syndrome is detected and an appropriate correction is made by applying gates one by one reverse order and initial state will be reproduced.
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