Kraus Operators for Single-Qubit Noise Channels

Recommended resources

The freely available resources this lesson will direct you to.

Introduction

Welcome to the first lesson in our deep dive into recent advances in quantum computing. We begin this journey in the "Quantum Computing Foundations" module by confronting a crucial aspect of real-world quantum systems: noise.

So far, quantum mechanics is often taught in the context of closed systems, which evolve perfectly according to the Schrödinger equation, represented by unitary transformations. However, any real quantum computer is an open system that inevitably interacts with its environment. This interaction introduces errors, or noise, which corrupts the quantum state and is the primary obstacle to building large-scale, fault-tolerant quantum computers.

In this lesson, we will build the mathematical framework to describe these noisy processes. Your background in theoretical physics and modeling astrophysical plasmas will be advantageous here. The core idea of treating a system of interest as a subsystem of a larger universe and tracing out the environmental degrees of freedom to obtain an effective, non-unitary dynamics is a concept central to many fields of physics.

Our goal is to derive the Kraus operator representation for standard single-qubit noise channels: bit-flip, phase-flip, and depolarizing. This formalism, also known as the operator-sum representation, is the cornerstone for describing any physically permissible evolution of a quantum state, including noise.

The Open System Viewpoint: From Unitary Evolution to Quantum Channels

The foundational principle for modeling the dynamics of an open quantum system is often called the "Church of the Larger Hilbert Space." It postulates that any open system S can be considered part of a larger, closed composite system S+E (where E is the environment or "bath"), which evolves unitarily. The non-unitary, noisy evolution of our system S is then found by "ignoring" the environment's state, which in mathematical terms means performing a partial trace over the environment's degrees of freedom.

This process gives us a map, called a quantum channel or a trace-preserving completely positive (TPCP) map, which describes the evolution of the system's density matrix. Let's see how this is derived.

IQIS Lecture 7.5 — Kraus representations

This video by Prof. Artur Ekert provides a concise derivation of the general form for the evolution of an open system. It starts from the physical picture of a unitary evolution on a combined system-environment state and formally shows how tracing out the environment leads to the operator-sum representation.

Please watch from the beginning until 06:29. Focus on how a unitary evolution on a combined system-environment state, U(ρ_S ⊗ ρ_E)U†, leads to the operator-sum representation, E(ρ_S) = Σ_k E_k ρ_S E_k†, after tracing out the environment.

As the video derives, the evolution of the system's density matrix, $ρ_{S}$ , can be expressed as:

ρ_{S} (t) = E (ρ_{S} (0)) = k \sum E_{k} ρ_{S} (0) E_{k}^{†}

This is the operator-sum representation, and the operators $E_{k}$ are known as Kraus operators (or operation elements). They encapsulate the entire effect of the environment's interaction on the system.

For this map to be physical (i.e., to always map a valid density matrix to another valid density matrix), the Kraus operators must satisfy the completeness condition:

k \sum E_{k}^{†} E_{k} = I

This condition ensures that the trace of the density matrix is preserved ( $T r (E (ρ)) = T r (ρ)$ ), which corresponds to the conservation of total probability. Any map that can be written in this operator-sum form is guaranteed to be a valid quantum operation (a CPTP map).

Deriving Kraus Operators for Standard Noise Channels

Now, we will apply this formalism to derive the Kraus operators for the three most common models of single-qubit noise. These channels serve as the fundamental building blocks for understanding and mitigating errors in quantum computers.

We will use the excellent lecture notes from John Preskill's course at Caltech as our primary reference. These notes provide clear, physically motivated derivations for each channel.

Quantum Information Chapter 3

John Preskill's lecture notes provide rigorous and clear derivations for several key quantum channels. We will focus on the sections detailing the depolarizing and dephasing channels.

Please read Section 3.4 'Three quantum channels' (starts on p. 24 of the PDF). Focus on the subsections for the Depolarizing channel (3.4.1) and the Dephasing channel (3.4.2). For each channel, pay attention to the physical model and how the 'Unitary representation' (describing the interaction with an environment) leads to the 'Operator-sum representation' (the Kraus operators).

Now, let's walk through the derivation for each channel, starting with the simplest case.

1. Bit-Flip Channel

The bit-flip channel is a model for errors where a qubit's state flips from $∣0 ⟩$ to $∣1 ⟩$ or vice versa.

Physical Model: With probability $p$ , a Pauli $X$ error ( $σ_{1}$ ) occurs. With probability $1 - p$ , nothing happens (an Identity operation, $I$ ).
Action on Density Matrix: We can write the evolution of an initial state $ρ$ as a weighted average of the outcomes:
$E (ρ) = (1 - p) I ρ I^{†} + pXρ X^{†}$
Kraus Operators: This expression is already in the operator-sum form. We can directly identify the Kraus operators:
- $E_{0} = 1 - p I = 1 - p (1001)$
- $E_{1} = p X = p (0110)$
Completeness Check: Let's verify the completeness condition:
$k \sum E_{k}^{†} E_{k} = E_{0}^{†} E_{0} + E_{1}^{†} E_{1} = (1 - p I)^{†} (1 - p I) + (p X)^{†} (p X)$ $= (1 - p) I^{2} + p X^{2} = (1 - p) I + p I = I$
The condition holds, as $X^{2} = I$ .

2. Phase-Flip (Dephasing) Channel

This channel models the loss of quantum coherence without loss of energy. It suppresses the off-diagonal elements of the density matrix in the computational basis. This is a very common type of noise in many physical qubit implementations.

Physical Model: With probability $p$ , a Pauli $Z$ error ( $σ_{3}$ ) occurs, introducing a phase flip for the $∣1 ⟩$ state. With probability $1 - p$ , nothing happens.
Action on Density Matrix:
$E (ρ) = (1 - p) I ρ I^{†} + pZρ Z^{†}$
Kraus Operators:
- $E_{0} = 1 - p I = 1 - p (1001)$
- $E_{1} = p Z = p (10 0 - 1)$
Completeness Check:
$k \sum E_{k}^{†} E_{k} = (1 - p) I + p Z^{2} = (1 - p) I + p I = I$
The condition holds, as $Z^{2} = I$ . This channel corresponds to the "Dephasing channel" in Preskill's notes. He uses a slightly different parameterization but the resulting map is the same.

3. Depolarizing Channel

The depolarizing channel is a symmetric noise model where the qubit state has some probability of being scrambled into any of the Pauli error states.

Physical Model: As described in Preskill's notes, with probability $1 - p$ , the state is unaffected. With probability $p$ , an error occurs, which is equally likely to be an $X$ , $Y$ , or $Z$ error. This means each Pauli error ${X, Y, Z}$ is applied with probability $p /3$ .
Action on Density Matrix: $E (ρ) = (1 - p) I ρ I^{†} + \frac{p}{3} Xρ X^{†} + \frac{p}{3} Y ρ Y^{†} + \frac{p}{3} Zρ Z^{†}$
Kraus Operators:
- $E_{0} = 1 - p I$
- $E_{1} = p /3 X$
- $E_{2} = p /3 Y$
- $E_{3} = p /3 Z$
Completeness Check: $k \sum E_{k}^{†} E_{k} = E_{0}^{†} E_{0} + E_{1}^{†} E_{1} + E_{2}^{†} E_{2} + E_{3}^{†} E_{3}$ $= (1 - p) I + \frac{p}{3} X^{2} + \frac{p}{3} Y^{2} + \frac{p}{3} Z^{2}$ $= (1 - p) I + \frac{p}{3} (I + I + I) = (1 - p) I + p I = I$ The condition holds, as $X^{2} = Y^{2} = Z^{2} = I$ .

Test your understanding!

An alternative, and very common, model for the depolarizing channel is one where with probability $p_{n e w}$ , the original state $ρ$ is discarded and replaced by the maximally mixed state $\frac{I}{2}$ , and with probability $1 - p_{n e w}$ it is left unchanged. The map is:
$E (ρ) = (1 - p_{n e w}) ρ + p_{n e w} \frac{I}{2}$

Show that this is equivalent to the first model of the depolarizing channel we discussed, and find the relationship between the error probability $p$ from our first model and $p_{n e w}$ in this model.

Hint: Use the identity $\frac{1}{3} (XρX + Y ρ Y + ZρZ) = \frac{4}{3} \frac{I}{2} - \frac{1}{3} ρ$ . This can be derived by expanding $ρ = \frac{1}{2} (I + P \cdot σ)$ and seeing how the Pauli matrices act on $σ$ .

Show answer

Let's start with the first model of the depolarizing channel:
$E (ρ) = (1 - p) ρ + \frac{p}{3} (XρX + Y ρ Y + ZρZ)$
Now, substitute the identity from the hint:
$E (ρ) = (1 - p) ρ + \frac{p}{3} (\frac{4}{3} \frac{I}{2} - \frac{1}{3} ρ)$
This is a typo in the hint, the correct identity is $XρX + Y ρ Y + ZρZ = (2 I - ρ)$ if Tr(ρ)=1... Wait, let's re-derive this.
Any single-qubit density matrix can be written $ρ = \frac{1}{2} (I + P \cdot σ)$ .
Let's see how $σ_{i} ρ σ_{i}$ transforms $P$ . For example, $X (P \cdot σ) X = P_{x} (X I X) + P_{y} (X Y X) + P_{z} (XZX) = P_{x} I - P_{y} Y - P_{z} Z$ .
So, $XρX = \frac{1}{2} (I + P_{x} σ_{x} - P_{y} σ_{y} - P_{z} σ_{z})$ .
Summing over $i \in {x, y, z}$ :
$XρX + Y ρ Y + ZρZ = \frac{1}{2} [3 I + (P_{x} - P_{x} - P_{x}) σ_{x} + (- P_{y} + P_{y} - P_{y}) σ_{y} + (- P_{z} - P_{z} + P_{z}) σ_{z}]$
$= \frac{1}{2} (3 I - P \cdot σ) = \frac{1}{2} (4 I - (I + P \cdot σ)) = 2 I - ρ$ .

Now, substituting this correct identity into the depolarizing channel map:
$E (ρ) = (1 - p) ρ + \frac{p}{3} (2 I - ρ)$
Let's rearrange the terms:
$E (ρ) = (1 - p - \frac{p}{3}) ρ + \frac{2 p}{3} I = (1 - \frac{4 p}{3}) ρ + \frac{4 p}{3} \frac{I}{2}$
This is now in the form $(1 - p_{n e w}) ρ + p_{n e w} \frac{I}{2}$ .
By comparing the two forms, we can see that they are equivalent if we set:
$p_{n e w} = \frac{4 p}{3}$
This equivalence is very useful. The first model is intuitive from an error-type perspective (what Pauli error happened?), while the second model gives a clear geometric picture: the Bloch sphere uniformly contracts towards the center. The factor $1 - 4 p /3$ is the amount by which the polarization vector shrinks.

Conclusion

In this lesson, we have established the fundamental formalism for describing noise in quantum systems. We have moved from the idealized world of closed, unitarily-evolving systems to the more realistic picture of open systems interacting with an environment.

Key Takeaways:

The evolution of an open quantum system can be described by a quantum channel ( $E$ ), which is a trace-preserving, completely positive (TPCP) map.
Any such channel can be expressed in the operator-sum representation using a set of Kraus operators ${E_{k}}$ : $E (ρ) = \sum_{k} E_{k} ρ E_{k}^{†}$ .
The Kraus operators must satisfy the completeness condition $\sum_{k} E_{k}^{†} E_{k} = I$ to conserve probability.
We derived the Kraus operators for three canonical single-qubit noise models:
- Bit-Flip: $E_{0} = 1 - p I, E_{1} = p X$
- Phase-Flip: $E_{0} = 1 - p I, E_{1} = p Z$
- Depolarizing: $E_{0} = 1 - p I, E_{1} = p /3 X, E_{2} = p /3 Y, E_{3} = p /3 Z$

This framework is not just a theoretical curiosity; it is the language used to characterize, simulate, and ultimately combat noise on real quantum hardware.

Preview of the next lesson:

Now that we have a tool to describe noise, we'll need a way to visualize its effect. In the next lesson, we will focus on single-qubit gates and their representation as rotations on the Bloch sphere. We will then explore how noise channels, like the ones we've just derived, affect a qubit's state vector on this sphere, providing a powerful geometric intuition for decoherence.