Quantum Error Correction Protects Quantum Search Algorithms Against Decoherence

Abstract

When quantum computing becomes a wide-spread commercial reality, Quantum Search Algorithms (QSA) and especially Grover’s QSA will inevitably be one of their main applications, constituting their cornerstone. Most of the literature assumes that the quantum circuits are free from decoherence. Practically, decoherence will remain unavoidable as is the Gaussian noise of classic circuits imposed by the Brownian motion of electrons, hence it may have to be mitigated. In this contribution, we investigate the effect of quantum noise on the performance of QSAs, in terms of their success probability as a function of the database size to be searched, when decoherence is modelled by depolarizing channels’ deleterious effects imposed on the quantum gates. Moreover, we employ quantum error correction codes for limiting the effects of quantum noise and for correcting quantum flips. More specifically, we demonstrate that, when we search for a single solution in a database having 4096 entries using Grover’s QSA at an aggressive depolarizing probability of 10−3, the success probability of the search is 0.22 when no quantum coding is used, which is improved to 0.96 when Steane’s quantum error correction code is employed. Finally, apart from Steane’s code, the employment of Quantum Bose-Chaudhuri-Hocquenghem (QBCH) codes is also considered.

Quantum Topological Error Correction Codes are Capable of Improving the Performance of Clifford Gates

“…The employment of QSCs for protecting the quantum circuits indeed increases the reliability of quantum computing, as shown in [61]. However, in order to avoid a perpetual encoding and decoding process throughout the quantum circuit before the number of errors exceeds the error correction capability of the QSC, we present a more efficient framework, where we encode the logical qubits at the input of the quantum circuit and decode them afterwards at the output of In the physical implementation of (a), we can imagine two layers of physical qubits, where on each layer the physical qubits are arranged over a lattice structure portrayed in Fig.…”

“…However, for colour codes [34], which belong to the QTECCs family, they can also be classified further as the member of a more specific category of quantum CSS codes, namely, the dual-containing CSS codes. For dual-containing CSS codes, a specific technique can be invoked for creating the associated quantum encoder V. This method of generating the quantum encoder V of dual-containing CSS codes has been detailed in [10], [61]. It is important to note that upon using the method detailed in [10], [57]- [59], [61], the number of quantum gates required to construct a quantum encoder V for a QSC C is linearly proportional to the number of physical qubits [57], [58].…”

Quantum Search Algorithms for Wireless Communications

“…In fact, a quantum annealing chipset [14] is already commercially available from D-Wave 1 [15], [16]. Apart from the quantum annealing architecture, the so-called gate-based architecture [10], which relies on building computational blocks using quantum gates in a similar fashion to classical logic gates, is attracting increasing attention due to the recent advances in quantum stabilizer codes [17]- [22], which are capable of mitigating the decoherence 2 effects encountered by quantum circuits [9]. In terms of implementation, D-Wave's most recent model, namely D-Wave 2000Q 3 , has a total of 2000 qubits, while IBM Q Experience 4 , which relies on the gated-based architecture, has currently only 20 qubits in total.…”

“…The channel envelope of each of the four paths may be deemed to fade independently. Assuming that the channel envelope at each path is quasi-static 22 during the channel estimation process, we may either estimate the four time-domain (TD) channel gains of the four paths, or the 512 frequency-domain (FD) subcarrier gains, which represent the Fast Fourier Transform (FFT) of the time-domain PDP, having taken the delay spread of the channel and the sampling frequency into consideration. Typically the FD channel is represented by the terminology of FD CHannel Transfer Function factor (FD-CHTF) [71].…”

Mitigation of Decoherence-Induced Quantum-Bit Errors and Quantum-Gate Errors Using Steane’s Code

“…This has the drawback that unknown information cannot be encoded. However, encoding unknown information is necessary because in many practical schemes QECC decoding and re-encoding are applied mid-way through the computation, as demonstrated in [4].…”