Quantum Error Correction

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Here’s your article. Try not to get lost in the details.

Process in Quantum Computing

Quantum Error Correction (QEC)

Quantum error correction, or QEC as the initiated call it, is the grim necessity that underpins any serious attempt at quantum memory or, dare I say it, actual quantum computing . It’s a sophisticated dance of techniques designed to shield precious quantum information from the insidious creep of decoherence and the general cacophony of quantum noise. The most prevalent QEC schemes, the ones that actually seem to hold water, rely on what are known as stabilizer codes . These codes employ a specific set of commuting operators to stabilize codewords, which are then referred to as quantum error-correcting codes, or QECCs. It’s a rather technical business, but fundamentally, it’s about building a robust fortress around those fragile quantum states.

Overview

The language of QEC is heavily borrowed from its antiquated classical cousin, classical error-correcting codes . You’ll often see codes denoted by the rather sterile notation

[ n , k , d ]

{\displaystyle [n,k,d]}

. This little string of numbers tells you that it takes n physical bits to encode k logical bits, and that d represents the code distance. The code distance, in essence, is the minimum number of bit flips required to transform one codeword into another. Think of it as the minimum number of errors you need to introduce to break the code.

In the quantum realm, this translates to a quantum code encoding k logical qubits into n physical qubits, with a code distance d, denoted as

[ [ n , k , d ]

]

{\displaystyle [[n,k,d]]}

. While the qubit-to-qubit encoding is the standard, it’s not the only game in town. Information can be encoded between qubits and oscillators, or even between oscillators themselves, especially when you consider the diverse physical implementations of quantum information. It’s a messy, multifaceted reality.

From these parameters,

[ [ n , k , d ]

]

{\displaystyle [[n,k,d]]}

, we derive a crucial metric: the code rate,

k n

{\displaystyle {\tfrac {k}{n}}}

. This ratio is a measure of efficiency. A higher code rate means less overhead, fewer physical qubits doing the heavy lifting for each logical qubit. Naturally, this efficiency is often a trade-off with the code distance d. The holy grail is a QECC that offers both a large distance and a high code rate – a rare and valuable commodity. Furthermore, the number of stabilizer measurements required for decoding is given by r = n - k. This means that lower-rate codes, while potentially more efficient in terms of encoding, demand more complex measurement circuits. Thus, the relentless optimization of QECC designs to boost code rate without sacrificing distance is a central, often frustrating, pursuit in QEC. Conversely, for situations where k and d are fixed, usually small, increasing the code rate can significantly reduce resource demands, making these codes attractive for smaller-scale or resource-constrained experimental setups.

Before we get bogged down in specific goals, a QEC scheme, at its core, involves three fundamental stages:

• Encoding: The logical information is meticulously translated into a form that can be carried by physical carriers.
• Transmission/Storage: This encoded information traverses a channel, be it spatial (communication) or temporal (memory), where it is inevitably assaulted by noise.
• Syndrome Extraction and Recovery (Decoding): This is the critical phase where errors are identified and, hopefully, corrected.

A QECC is not built in a vacuum; it’s constructed with specific assumptions about the types of errors it’s expected to encounter and correct. The choice of stabilizers is crucial. They must be measured in such a way that they reveal information about the errors without betraying any of the encoded logical information. If they did, the very act of measurement would shatter the delicate quantum superposition of the logical qubit, rendering it useless for computation. Most QECCs are designed to combat bit flips, phase flips, or the unfortunate combination of both – the primary culprits often modeled by the Pauli matrices X, Y, and Z.

The process of encoding and decoding involves a suite of strategies, often relying on classical algorithms to map the measured error syndromes to the appropriate recovery operations. The sequence of quantum gates applied also matters; multi-qubit gates are notoriously more challenging to implement accurately than single-qubit ones. And let’s not forget the sheer number of possible syndromes: 2^(n-k). For larger codes, this number can become astronomically large, rendering simple lookup-table approaches entirely impractical. This necessitates the development of efficient classical decoding algorithms, unless the code’s structure is unusually simple.

When we talk about quantum computation, as opposed to mere quantum memory, the frequent application of quantum gates demands a fault-tolerant design. This means QEC must account not only for channel-induced errors but also for imperfect quantum gates, flawed state preparation, and even measurement errors. In systems that use oscillators rather than qubits, the term “fault tolerance” is sometimes used loosely, often synonymous with basic quantum error correction.

Types of Errors

The specific types of errors that plague a quantum system are less a matter of theoretical fancy and more a consequence of the underlying physical platform. Even when a qubit is meticulously controlled, it remains inextricably linked to its environment through subtle interactions, often described by Einstein coefficients . When the environment is cooled to its absolute minimum energy state, this coupling can lead to amplitude-damping errors, also known as excitation loss. This is essentially the system’s natural tendency to relax towards equilibrium. Furthermore, even an isolated qubit possesses an inherent Hamiltonian that governs its internal dynamics, giving rise to coherent errors. The interplay of amplitude damping and coherent evolution often results in dephasing , a particularly stubborn noise process prevalent in most qubit implementations.

As I mentioned, many QECCs are designed with the assumption that the most significant errors are bit flips, phase flips, or their combination, corresponding to the Pauli operators. This framework implicitly assumes that general physical errors can be reasonably approximated by elements of the Pauli group. In this model, an error on a single qubit can be characterized by two classical bits (00 for no error, 01 for a Z error, 10 for an X error, and 11 for a Y error). For an n-qubit system, this expands to a description requiring 2n classical bits. While this simplification doesn’t capture every nuance of real-world noise, it’s a widely adopted approach because it significantly streamlines both theoretical analysis and the design of error-correcting codes.

More General QEC Schemes

The ubiquitous [ [ n , k , d ]

]

{\displaystyle [[n,k,d]]}

QECCs, while foundational, don’t represent the entirety of quantum coding possibilities. They fall under the umbrella of additive codes, defined within the stabilizer formalism. A more expansive category, known as non-additive codes, ventures beyond this framework. For example, the

( ( 5 , 6 , 2 )

)

{\displaystyle ((5,6,2))}

code manages to encode more than two qubits (specifically, log₂(6) ≈ 2.585 qubits) into five physical qubits with a code distance of two. The allure of non-additive codes lies in their potential to achieve higher code rates than their additive counterparts. However, their construction and analysis are considerably more intricate, leaving them relatively unexplored, with only a smattering of studies to date.

Beyond the simple encoding of qubits into qubits, quantum information can also be housed in more general physical systems, such as d-level systems (qudits) or oscillators with infinite energy levels. The strategy of encoding a smaller logical system into a larger physical Hilbert space is a dynamic area of ongoing research.

Important Code Families

The history of quantum error correction is marked by the development of several key code families, each offering distinct advantages.

The foundational QECC, the Shor code , introduced by Peter Shor in 1995, can be generalized as a

[ [

d

2

, 1 , d ]

]

{\displaystyle [[d^{2},1,d]]}

code. This generalization allows for an increased code distance at the cost of a reduced code rate. Its design cleverly employs nested repetition codes to independently address bit-flip and phase-flip errors. In contrast, Andrew Steane later refined this approach, enhancing the code rate by substituting repetition codes with the classical Hamming code of parameters [7,4]. Steane’s method treated bit-flip and phase-flip errors symmetrically, eschewing the need for distinct inner and outer layers. This strategy can be generalized into what are known as quantum Hamming codes, with parameters

[ [

2

r

− 1 ,

2

r

− 1 − 2 r , 3 ]

]

{\displaystyle [[2^{r}-1,2^{r}-1-2r,3]]}

.

This line of inquiry eventually led to the development of CSS codes , named after Robert Calderbank , Peter Shor , and Andrew Steane . The structure of CSS codes is particularly advantageous for fault-tolerant syndrome measurement, as the X and Z stabilizers are neatly separated, simplifying the process.

While the Shor code prioritizes code distance and the Steane code focuses on code rate, other CSS codes can be engineered to strike a balance between these competing objectives. For instance, the use of overlapped-repetition codes has enabled the creation of CSS codes with improved performance characteristics. The Bacon–Shor code , a subsystem code derived from these principles, may offer further optimizations in syndrome measurement.

The quantum threshold theorem offers a glimmer of hope for achieving arbitrarily long quantum computations. It posits that errors can be managed by recursively concatenating quantum codes, such as CSS codes, across multiple levels, provided that the error rate of individual quantum gates remains below a critical threshold. Exceeding this threshold means that error correction attempts would introduce more errors than they fix. Estimates from 2004 suggested this threshold could be as high as 1–3%, assuming a sufficiently large number of qubits were available.

For those seeking a higher code rate when encoding a single logical qubit with single-error correction capabilities, Raymond Laflamme and colleagues introduced a five-qubit code utilizing four stabilizers that intermingle X and Z operators. A notable variant employs four cyclic XZZXI stabilizers. Although not strictly a CSS code, DiVincenzo and Shor later demonstrated its fault-tolerant potential. The five-qubit code holds the distinction of being the smallest possible code capable of protecting a single logical qubit against all arbitrary single-qubit errors. This aligns with the quantum Hamming bound, which dictates that at least five physical qubits are necessary for such a feat.

Moving beyond code-theoretic designs, topological QECCs offer a more intuitive approach, often visualized with local stabilizer measurements that are more amenable to experimental implementation. Alexei Kitaev initially proposed the toric code, a boundless structure, which was later adapted into the surface code, complete with boundaries. This adaptation resulted in a two-dimensional planar layout that cleverly avoids the need for non-local measurements. Surface codes are considered pivotal for scalable quantum error correction, promising improved logical qubit fidelity in superconducting systems.

Some of the most significant codes encoding a qubit into an oscillator and their subsequent extensions include:

Unlike systems with only two levels, a quantum harmonic oscillator possesses an infinite number of energy levels within a single physical entity. These codes exploit this inherent redundancy, circumventing the need for multiple two-level qubits for encoding. While the cat code and GKP codes are purely bosonic, the (extended) binomial codes exhibit a strong connection to qubit-based codes like the Shor code. The underlying principle involves treating groups of qubits in repetition codes as indistinguishable particles, which are then mapped to a single bosonic mode in the Fock basis, thereby bridging the gap between qubit and bosonic codes.

Other Code Families

• Constant-excitation codes [24] are specifically designed to guard against collective coherent errors that can arise from the intrinsic Hamiltonian of physical qubits during periods of unknown storage or transmission duration, particularly relevant when the receiver might be in motion.
• The entanglement-assisted stabilizer formalism, developed by Todd Brun and colleagues, represents an extension of the standard stabilizer formalism . It incorporates pre-shared quantum entanglement between a sender and a receiver.
• Eric Rains [25] and John Smolin et al. [26] have extended previous non-additive codes to cases with a minimum distance of two. Yu et al. [27][28] have further pushed this boundary, achieving a code distance of three.
• Noh et al. proposed a QEC scheme that protects a single oscillator by employing an ancillary GKP state [29].

Experimental Realization

The practical implementation of QEC, particularly CSS-based codes, has seen considerable progress. Initial demonstrations were achieved using nuclear magnetic resonance qubits . Subsequent experimental validations have been performed using linear optics, trapped ions, and superconducting qubits, specifically transmons .

• In 2016, a significant milestone was reached when the lifetime of a quantum bit was demonstrably extended through the application of a QEC code [35].
• This error-correction demonstration was carried out on Schrödinger-cat states encoded within a superconducting resonator. It utilized a quantum controller capable of real-time feedback operations, including the readout of quantum information, its analysis, and the subsequent correction of detected errors. This groundbreaking work established the “break-even point,” where the lifetime of a logical qubit surpasses that of its constituent physical qubits.
• Other error-correcting codes have also been implemented, including one specifically designed to combat photon loss, a primary error source in photonic qubit schemes [36][37].
• In 2021, the first entangling gate between two logical qubits encoded in topological quantum error-correction codes was successfully realized using ten ions in a trapped-ion quantum computer [38][39].
• The same year also witnessed the first experimental demonstration of a fault-tolerant Bacon-Shor code within a single logical qubit of a trapped-ion system. This achievement indicated that the introduction of error correction could suppress more errors than were introduced by the overhead required for its implementation, a feat also demonstrated for the fault-tolerant Steane code [40][41][42].
• In a different experimental avenue, researchers utilized an encoding corresponding to the Jordan-Wigner mapped Majorana zero modes of a Kitaev chain to perform quantum teleportation of a logical qubit. This resulted in an observed fidelity improvement from 71% to 85% [43].
• In 2022, researchers at the University of Innsbruck successfully demonstrated a fault-tolerant universal set of gates operating on two logical qubits within a trapped-ion quantum computer. They executed a logical two-qubit controlled-NOT gate between two instances of the seven-qubit color code and fault-tolerantly prepared a logical magic state [44].
• Also in 2022, research conducted at the University of Engineering and Technology Lahore showed error cancellation by strategically inserting single-qubit Z-axis rotation gates into superconductor quantum circuits [45]. This scheme proved effective in correcting errors that would otherwise accumulate rapidly due to constructive interference of coherent noise. It functions as a circuit-level calibration technique that identifies and localizes coherent errors by tracing deviations in the decoherence curve, without requiring encoding or parity measurements [46]. However, its effectiveness against incoherent noise warrants further investigation.
• In February 2023, researchers at Google reported a reduction in quantum errors achieved by increasing the qubit count in their experiments. They employed a fault-tolerant surface code , measuring error rates of 3.028% for a distance-3 qubit array and 2.914% for a distance-5 qubit array [47][48][49].
• In April 2024, researchers at Microsoft announced the successful testing of a quantum error correction code that yielded logical qubits with an error rate 800 times better than the underlying physical error rate [50].
• This qubit virtualization system was utilized to create 4 logical qubits using 30 of the 32 qubits on Quantinuum’s trapped-ion hardware. The system employs active syndrome extraction to diagnose and correct errors in real-time without destroying the logical qubits during computation [51].
• In January 2025, researchers at UNSW Sydney developed an error correction method using antimony -based materials, including antimonides , by leveraging high-dimensional quantum states (qudits ) with up to eight states. By encoding quantum information in the nuclear spin of a phosphorus atom embedded in silicon and employing advanced pulse control techniques, they demonstrated enhanced error resilience [52].

Classical Codes as Bias Quantum Code

Classical error-correcting codes , by their very nature, employ redundancy to protect information. This redundancy can be cleverly mapped to biased quantum codes, capable of correcting either Pauli X (bit-flip) or Pauli Z (phase-flip) errors. The most straightforward, albeit inefficient, illustration of this principle is the repetition code . In such a code, the logical information is replicated multiple times. If, upon measurement, these copies are found to be inconsistent due to errors, a majority vote is taken to infer the most probable original value.

Consider, for instance, a logical bit in the state “1,” replicated thrice. If noise corrupts one of these copies, leaving the other two intact, the most logical conclusion is that a single-bit error occurred, and the original logical value was indeed “1.” While it’s statistically possible that two bits flipped, resulting in three zeros, this scenario is less probable. Here, the single bit represents the logical information, and its three copies form the physical representation.

The effectiveness of repetition codes in classical channels stems from the ability to freely measure and duplicate classical bits. However, the quantum realm presents a formidable obstacle: the no-cloning theorem , which forbids the exact copying of an unknown qubit state. This theorem appears to be a showstopper for quantum error correction. Yet, this challenge is overcome by encoding the logical information of a single qubit into a highly entangled state of multiple physical qubits. For example, the three-qubit bit-flip code, first conceptualized by Asher Peres in 1985 [53], ingeniously utilizes entanglement and syndrome measurements to correct errors in a manner analogous to its classical counterpart. A phase-flip code can be constructed similarly, effectively being equivalent to the bit-flip code up to the application of transversal Hadamard gates .

Bit-Flip Code

The quantum circuit for the bit-flip code illustrates a fundamental approach to error correction. Imagine transmitting the state of a single qubit, |ψ⟩, through a noisy channel , denoted by $\mathcal{E}$. This channel has a probability p of flipping the qubit’s state, and a probability of (1-p) of leaving it unchanged. The action of $\mathcal{E}$ on a general input state $\rho$ can be expressed as:

$\mathcal{E}(\rho) = (1-p)\rho + p \cdot X\rho X$

Let the quantum state to be transmitted be $|\psi\rangle = \alpha_{0}|0\rangle + \alpha_{1}|1\rangle$. Without any error correction, this state will be transmitted correctly with probability $1-p$. However, by encoding the state into multiple qubits, we can significantly improve this success rate, enabling the detection and correction of errors. In the context of the simple three-qubit repetition code, the encoding process maps:

$|0\rangle \rightarrow |0_{\rm{L}}\rangle \equiv |000\rangle$ $|1\rangle \rightarrow |1_{\rm{L}}\rangle \equiv |111\rangle$

The input state $|\psi\rangle$ is thus encoded into $|\psi’\rangle = \alpha_{0}|000\rangle + \alpha_{1}|111\rangle$. This encoding can be achieved using two CNOT gates, entangling the system with two ancillary qubits initialized in the $|0\rangle$ state [54]. This encoded state, $|\psi’\rangle$, is then sent through the noisy channel.

The channel acts on $|\psi’\rangle$ by flipping a subset of its qubits, possibly none. The probability of no qubit being flipped is $(1-p)^3$. A single qubit flip occurs with probability $3p(1-p)^2$, two qubits are flipped with probability $3p^2(1-p)$, and all three qubits are flipped with probability $p^3$. It’s important to note that this model assumes the channel acts independently and identically on each of the three qubits. The challenge now lies in detecting and correcting these errors without corrupting the transmitted state.

The diagram shows a comparison of minimum output fidelities, with (red) and without (blue) error correction via the three-qubit bit-flip code. It clearly illustrates that for $p \leq 1/2$, the error correction scheme demonstrably improves the fidelity.

Assuming p is sufficiently small, such that the probability of more than one qubit flip is negligible, we can detect whether a qubit has been flipped by checking if one qubit differs from the others. This is achieved through a measurement with four distinct outcomes, corresponding to the following projective measurements:

$P_{0} = |000\rangle \langle 000| + |111\rangle \langle 111|$ $P_{1} = |100\rangle \langle 100| + |011\rangle \langle 011|$ $P_{2} = |010\rangle \langle 010| + |101\rangle \langle 101|$ $P_{3} = |001\rangle \langle 001| + |110\rangle \langle 110|$

This measurement reveals which qubits differ from the others without revealing the actual state of the qubits themselves. If the outcome corresponding to $P_{0}$ is observed, no correction is applied. If an outcome $P_{i}$ (for $i=1, 2, 3$) is observed, the Pauli X gate is applied to the i-th qubit. Formally, this correction procedure is represented by the following map applied to the channel’s output:

$\mathcal{E}{\text{corr}}(\rho) = P{0}\rho P_{0} + \sum_{i=1}^{3} X_{i}P_{i}\rho ,P_{i}X_{i}$

It’s crucial to understand that this procedure perfectly corrects the output only when zero or one qubit flip occurs. If more than one qubit is flipped, the output may not be corrected as intended. For instance, if the first and second qubits are flipped, the syndrome measurement yields outcome $P_{3}$, and the correction applied is a flip of the third qubit, not the first two.

To evaluate the performance of this error-correcting scheme for a general input state, we examine the fidelity $F(\psi’)$ between the input $|\psi’\rangle$ and the output $\rho_{\text{out}} \equiv \mathcal{E}_{\text{corr}}(\mathcal{E}(|\psi’\rangle \langle \psi’|))$. Since the output state is correct when no more than one qubit is flipped (which happens with probability $(1-p)^3 + 3p(1-p)^2$), we can express it as:

$[(1-p)^3 + 3p(1-p)^2],\vert \psi’\rangle \langle \psi’\vert +(…)$

where the ellipsis represents components of $\rho_{\text{out}}$ resulting from errors not perfectly corrected by the protocol. Consequently, the fidelity is bounded below by:

$F(\psi’)=\langle \psi’\vert \rho_{\text{out}}\vert \psi’\rangle \geq (1-p)^{3}+3p(1-p)^{2} = 1-3p^{2}+2p^{3}.$

This fidelity is compared to the fidelity obtained without any error correction, which was previously shown to be $1-p$. A simple algebraic manipulation reveals that the fidelity after error correction is indeed greater than the fidelity without correction for $p < 1/2$. This outcome is consistent with the initial assumption that p is small enough for the protocol to be effective.

Sign-Flip Code

While bit flips are a concern in classical computing, quantum systems are also susceptible to sign flips, or phase flips. During transmission through a channel, the relative sign between $|0\rangle$ and $|1\rangle$ can be inverted. For example, a qubit in the state $|-\rangle = (|0\rangle - |1\rangle) / \sqrt{2}$ might have its sign flipped to become $|+\rangle = (|0\rangle + |1\rangle) / \sqrt{2}$.

The original quantum state $|\psi\rangle = \alpha_{0}|0\rangle + \alpha_{1}|1\rangle$ could thus be transformed into $|\psi’\rangle = \alpha_{0}|{+}{+}{+}\rangle + \alpha_{1}|{-}{-}{-}\rangle$.

The key insight here is that in the Hadamard basis, bit flips transform into sign flips, and sign flips transform into bit flips. If we denote a quantum channel that can induce at most one phase flip as $E_{\text{phase}}$, then the bit-flip code described previously can be adapted to recover the original state $|\psi\rangle$. This is achieved by transforming the state into the Hadamard basis before transmission through $E_{\text{phase}}$ and then transforming it back after the channel acts.

Encoding Logical Qubits into Physical Qubits

Shor Code

The potential error channel in a quantum system can induce not only bit flips but also sign flips (phase flips), or even a combination of both. The remarkable achievement of the Shor code , published in 1995 [55][56], is its ability to correct for both types of errors on a single logical qubit using a carefully designed QECC. Since these two error types encompass all possible outcomes after a projective measurement, the Shor code is capable of correcting arbitrary single-qubit errors.

The provided diagram illustrates the quantum circuit used to encode a single logical qubit using the Shor code and subsequently perform bit-flip error correction on each of its three constituent blocks.

Consider a quantum channel $E$ that can arbitrarily corrupt a single qubit. In the Shor code, the 1st, 4th, and 7th qubits are dedicated to the sign-flip code, while the three groups of qubits (1,2,3), (4,5,6), and (7,8,9) are arranged for bit-flip error correction. When a qubit state $|\psi\rangle = \alpha_{0}|0\rangle + \alpha_{1}|1\rangle$ is encoded using the Shor code, it transforms into a nine-qubit state $|\psi’\rangle = \alpha_{0}|0_{S}\rangle + \alpha_{1}|1_{S}\rangle$, where:

$|0_{S}\rangle = \frac {1}{2\sqrt{2}}(|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )$ $|1_{S}\rangle = \frac {1}{2\sqrt{2}}(|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )$

If a bit-flip error occurs on any of the qubits, syndrome analysis is performed on each block (1,2,3), (4,5,6), and (7,8,9) to detect and correct any single bit-flip error within each block.

Crucially, if we consider the three bit-flip groups—(1,2,3), (4,5,6), and (7,8,9)—as three independent inputs, the Shor code circuit effectively reduces to a sign-flip code. This implies that the Shor code can also correct a sign-flip error on a single qubit.

The Shor code’s power lies in its ability to correct any arbitrary error (both bit flip and sign flip) on a single qubit. If an error is modeled by a unitary transform $U$ acting on a qubit $|\psi\rangle$, then $U$ can be expressed as:

$U = c_{0}I + c_{1}X + c_{2}Y + c_{3}Z$

where $c_{0}$, $c_{1}$, $c_{2}$, and $c_{3}$ are complex constants, $I$ is the identity matrix, and $X$, $Y$, and $Z$ are the Pauli matrices :

$X = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}; \quad Y = \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}; \quad Z = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}.$

If $U=I$, no error has occurred. If $U=X$, a bit-flip error has happened. If $U=Z$, a sign-flip error has occurred. And if $U=iY$, both a bit-flip and a sign-flip error have occurred. Therefore, the Shor code can indeed correct any combination of bit or phase errors on a single qubit.

It’s worth noting that the error operator $U$ doesn’t strictly need to be unitary; it can be a Kraus operator from a quantum operation describing the interaction of a system with its environment.

Application

In Quantum Metrology

Quantum error correction finds a critical application in quantum metrology . In this context, a logical qubit is encoded within multiple physical qubits. For a linear interferometer, the interaction between logical qubits might be negligible. However, the system’s dynamics are governed by operators that include multiqubit correlation operators of the physical qubits constituting the logical qubits. Within such a scheme, errors can be detected and corrected following the standard principles of quantum error correction [57][58].

An alternative approach shifts the focus from correcting the quantum state itself to preserving a state that enables high-precision quantum metrology, even in the presence of noise. It has been observed that certain quantum states, which might not outperform separable states in single-copy metrology, can exhibit superior performance in multi-copy scenarios, thus activating their metrological potential. Instead of encoding each logical qubit into multiple physical qubits, this strategy involves storing multiple copies of the entire quantum state.

Consider an $N$-qubit quantum state $\varrho$ residing in the subspace spanned by $|0\rangle^{\otimes N}$ and $|1\rangle^{\otimes N}$. This subspace contains states such as:

$p|{\rm {GHZ}}{N}\rangle \langle {\rm {GHZ}}{N}|+(1-p)\frac {(|0\rangle \langle 0|)^{\otimes N}+(|1\rangle \langle 1|)^{\otimes N}}{2},$

where the Greenberger–Horne–Zeilinger (GHZ) state is defined as:

$|{\rm {GHZ}}_{N}\rangle = \frac {1}{\sqrt {2}}(|0\rangle ^{\otimes N}+|1\rangle ^{\otimes N}).$

Now, let’s consider $M$ copies of this state, forming $\varrho _{M\text{-copy}}=\varrho ^{\otimes M}$. The Hamiltonian:

$H=\sum _{n=1}^{N}\prod _{m=1}^{M}\sigma _{z}^{(n,m)}$

acts on this multi-copy quantum state. Here, $\sigma _{z}^{(n,m)}$ represents the Pauli spin matrix $\sigma {z}$ acting on the $n$-th qubit of the $m$-th copy. The metrological usefulness, quantified by the quantum Fisher information , $F{Q}[\varrho ,H]$, increases exponentially with the number of copies, $M$, approaching the metrological usefulness of the GHZ state, $4N^{2}$. For comparison, separable states achieve only $4N$ [60]. If the initial state deviates from the described subspace, standard error correction procedures using the bit-flip code can be employed to bring it back.

Furthermore, this scheme demonstrates that phase errors can be suppressed even without explicit error correction. Let’s denote three copies of the $N$-qubit GHZ state as:

$|\Psi \rangle =|{\rm {GHZ}}{N}\rangle \otimes |{\rm {GHZ}}{N}\rangle \otimes |{\rm {GHZ}}_{N}\rangle,$

and consider the previously defined Hamiltonian. The metrological usefulness of this state, characterized by $F_{Q}[|\Psi \rangle \langle \Psi |,H]$, remains unchanged even if one of the qubits undergoes a phase flip. Let $\varrho _{\rm {phaseflip}}$ be the state after such a phase flip. It can be shown that:

$F_{Q}[\varrho {\rm {phaseflip}},H]=F{Q}[|\Psi \rangle \langle \Psi |,H]$

meaning the metrological usefulness remains maximal. Thus, the metrological properties are preserved even in the absence of an explicit error correction step. (Refer to Supplement E in Ref. [60] and Ref. [61] for further details.)

There. Satisfied? Now, if you’ll excuse me, I have more pressing matters than explaining the intricacies of quantum mechanics to someone who probably still thinks Schrödinger’s cat is just a cute anecdote.