Unpacking the Paradox: A Deeper Look at Quantum Teleportation (part one)

11 min readApr 7, 2024

Teleporting qubits: artist’s impression a wormhole created in a quantum processor. (Courtesy: A Mueller/Caltech)

Have you ever dreamt of stepping onto a platform and instantly materializing on the other side of the world? From the iconic transporter beams of Star Trek to the D-Mails of Steins;Gate, teleportation has captured our imaginations for decades. But while science fiction paints at picture of beaming entire people across vast distances, the reality of quantum teleportation is far more fascinating, and fundamentally different.

Teleported “Gel-Banana” from Steins;Gate

The world of quantum mechanics can feel like a foreign land, filled with complex concepts and unfamiliar terminology. Stepping into this field can be terrifying, and rightfully so!

This blog post is designed to be digestible in manageable portions, without succumbing to information overload.

Take your time, read at your own pace, and feel free to revisit sections as needed. Let’s delve into the world of the “spooky actions at distances”, one qubit at a time.

The humble Qubit

In the world of classical computing, information is processed and stored in bits, which can exist in one of two states: 0 or 1. However, the quantum realm introduces us to a whole new level of complexity with the qubit, short for quantum bit. To wrap our heads around this concept, let’s come up with a thought experiment involving Schrödinger’s cat.

For Dummies: Schrödinger’s Cat Paradox explained — Cultura Colectiva

Imagine a box containing the following objects:

a Cat
a Vial of Poison
a Radioactive Atom.

According to quantum mechanics, until we open the box and observe its contents, the cat exists in a superposition of two states: both alive and dead simultaneously. This bizarre scenario illustrates the fundamental principle of superposition, where particles can exist in multiple states at once.

Classical bits with either 0 or 1 states on the left and a qubit on the right

Now, let’s apply this principle to the qubit. Unlike classical bits, which are confined to either a 0 or 1 state, a qubit can exist in a superposition of both states simultaneously. In essence, it’s as if our digital coin is simultaneously heads and tails until we flip it and observe the outcome.

Just as Schrödinger’s cat remains in a state of uncertainty until observed, the qubit’s state is uncertain until measured. This uncertainty is encapsulated by another fundamental concept of quantum mechanics: Heisenberg’s uncertainty principle, which dictates that certain pairs of physical properties, such as position and momentum, cannot be precisely measured simultaneously.

But, what is the point of Qubits? Why would I use it?

The essence of qubits lies in their ability to harness the principles of quantum mechanics to exponentially increase computational power. Unlike classical bits, which are confined to binary states, qubits can exist in superpositions of multiple states simultaneously, enabling quantum computers to perform vast numbers of calculations in parallel.

This parallelism, coupled with the phenomenon of quantum entanglement, allows for the creation of quantum algorithms that can solve certain problems exponentially faster than their classical counterparts. In essence, qubits represent a paradigm shift in computing, promising to revolutionize fields such as cryptography, optimization, and drug discovery by tackling complex problems that are currently beyond the reach of classical computers.

Alain Aspect, John Clauser, and Anton Zeilinger won the Physics Nobel Prize in 2022 for confirming the existence of Entanglement which defied predictions based on classical physics, including those made by Albert Einstein himself.

Ok, but how does it affect me, an average Joe?

Consider the problem of factoring large numbers, a task crucial for encryption methods like RSA. In classical computing, the best-known algorithm for factoring large numbers is the general number field sieve (GNFS), which becomes increasingly inefficient as numbers grow larger. For example, factoring a 300-digit number using GNFS could take millennia with current technology.

Shor-s-Algorithm/Shor.ipynb at master · mett29/Shor-s-Algorithm (github.com)

On the other hand, quantum computing offers a solution through Shor’s algorithm, which leverages the unique properties of qubits to factor large numbers exponentially faster than classical algorithms. While Shor’s algorithm may still be in its infancy due to the technological challenges of building large-scale quantum computers, theoretical analyses suggest that it could factor large numbers in polynomial time, dramatically reducing the computational burden compared to classical approaches to factor large numbers, thereby breaking the RSA encryption used to encrypt passwords!

What Roles Do |0⟩ and |1⟩ Characters Play in Quantum Diagrams?

In quantum diagrams, the characters |0⟩ (pronounced “Ket” 0) and |1⟩ (pronounced “Ket” 1) represent the basis states of a qubit in a computational basis using Dirac Notations. These states correspond to the classical binary states of 0 and 1, but in the quantum realm, they can exist simultaneously in superpositions.

The |0⟩ state signifies that the qubit is in the state where its probability amplitude for being measured as 0 is 1, and its amplitude for being measured as 1 is 0.
The |1⟩ state indicates the opposite scenario, with a probability amplitude of 1 for being measured as 1 and 0 for being measured as 0.

The concept of basis states extends beyond the computational basis to other representations, such as the eigenstates of observable properties like position, momentum, or spin. For example, in the context of spin, basis states might represent the spin of a particle along different axes, such as |↑⟩ and |↓⟩ for spin-up and spin-down along the z-axis. It’s quite similar to how the coordinates (X, Y and Z) works in classical coordinate systems!

All qubit states can be represented as a sum of their basis states.

Let’s take the following qubit as an example,

Vector representation of a generic qubit state

In this context, α² denotes the likelihood of the quantum state |ψ⟩ being measured as |0⟩, while β² represents the probability of it being measured as |1⟩.

It can also be written as the sum of the probabilities in the following notation:

Basis state representation of a generic qubit state

In addition to the “Ket”notation commonly used in quantum mechanics, there exists another notation known as the “Bra” notation. This notation is denoted by the symbol “⟨” (pronounced as “bra”) and represents the complex conjugate transpose of a ket vector.

While the ket notation represents quantum states as column vectors, the bra notation represents their dual or adjoint states as row vectors. Together, these notations form the basis for expressing quantum states and operations comprehensively.

In quantum mechanics, the bra notation, denoted by ⟨, and the ket notation, denoted by ⟩, can be combined to form what’s known as bra-ket notation. This combined notation, also referred to as Dirac notation after physicist Paul Dirac, allows for concise and elegant expressions of quantum states, operators, and measurements.

By pairing a bra vector with a ket vector, physicists can represent various quantum mechanical concepts, such as inner products, outer products, and expectation values, in a compact and intuitive manner.

Let’s consider a quantum state represented by the ket vector |ψ⟩, defined as:

|ψ⟩ = α|0⟩ + β|1⟩

In bra-ket notation, the corresponding bra vector for |ψ⟩ is ⟨ψ|, which is the complex conjugate transpose of |ψ⟩. Therefore:

⟨ψ| = α*⟨0| + β*⟨1|

Here, ⟨0| and ⟨1| represent the bra vectors corresponding to the basis states |0⟩ and |1⟩, respectively.

Linear algebra & transformation using bra-kets

What is the spherical representation of the Qubit, then?

Bloch spheres are a geometric representation used to visualize the state of a single qubit in quantum mechanics.

In the Bloch sphere model, the surface of a sphere represents all possible quantum states of the qubit. The north pole of the sphere corresponds to the state |0⟩, while the south pole corresponds to the state |1⟩. The equator of the sphere represents superpositions of |0⟩ and |1⟩, with different points on the equator representing different combinations of these states.

One of the key features of the Bloch sphere is its ability to visually represent quantum operations. Quantum gates, such as Pauli-X, Pauli-Y, and Pauli-Z gates, correspond to rotations of the qubit’s state vector on the surface of the sphere. These rotations can be visualized as movements along different axes of the sphere.

What are these X, Y and Z gates?

In the context of quantum computing, gates play a crucial role in manipulating qubits, the quantum equivalent of bits in classical computers. Unlike bits, which can only be 0 or 1, qubits can exist in a state called superposition, meaning they can be both 0 and 1 simultaneously. This opens up a whole new realm of possibilities for computation.

Quantum gates bear a striking resemblance to their classical counterparts, echoing the familiar logic gates used in classical computing. Just as classical gates manipulate bits to perform logical operations, quantum gates operate on qubits, the quantum equivalent of bits, to enact quantum transformations. However, while classical gates are deterministic, quantum gates harness the probabilistic nature of quantum mechanics to achieve unique computational capabilities.

These gates perform various operations on qubits, transforming their states in specific ways. Some common types of quantum gates include:

Quantum Gates Similar to Classical Gates:

Pauli-X Gate: Similar to classical NOT gate, flips the qubit from state |0⟩ to |1⟩ and vice versa.
Pauli-Z Gate: Resembles classical bitwise inversion, applies a phase flip to the qubit state.
Hadamard Gate: Creates superposition by transforming the basis states |0⟩ and |1⟩ into equal superpositions of both states.

Quantum Gates Not Similar to Classical Gates:

CNOT Gate: Performs an XOR operation on two qubits, flipping the target qubit if the control qubit is in state |1⟩.
Phase Gate: Introduces a phase shift to the qubit’s state, rotating it around the Z-axis of the Bloch sphere.
Toffoli Gate: Acts on three qubits and flips the target qubit if both control qubits are in state |1⟩.

How do I get the values of Qubits if they can be in multiple states at a time?

The answer lies in the process of quantum measurement. Despite their superposition, qubits yield definite outcomes when measured. However, the outcomes are probabilistic, governed by the principles of quantum mechanics.

To extract the values of qubits, we perform measurements that collapse their superpositions into one of the possible states. For instance, in the computational basis, a measurement will yield either the outcome |0⟩ or |1⟩, each with a probability determined by the coefficients of the qubit’s superposition. However, the process of measurement is not straightforward, owing to the principles of quantum mechanics and the no-cloning theorem.

The no-cloning theorem dictates that it is impossible to create an exact copy of an arbitrary unknown quantum state. This theorem fundamentally limits our ability to precisely replicate or clone quantum information, adding an extra layer of complexity to the measurement process.

Crucially, the act of measurement irreversibly alters the quantum state, revealing one aspect of the qubit’s nature while obscuring others. This irreversible transformation underscores the inherently probabilistic nature of quantum systems.

Moreover, measuring qubits is non-trivial due to the uncertainty principle, which dictates that certain pairs of physical properties, such as position and momentum, cannot be precisely measured simultaneously. As a result, extracting complete information about a qubit often requires multiple measurements and sophisticated techniques.

Measurement collapses a qubit into a classical bit

Errors and Quantum Error Correction

Probability of getting a logical error after decoding versus number of rounds run, shown for various sizes of phase-flip repetition code. (Demonstrating the Fundamentals of Quantum Error Correction — Google Research Blog)

Quantum systems are fragile, susceptible to various sources of noise and interference that can distort measurement outcomes. These errors can stem from environmental factors, imperfections in hardware, or limitations in measurement techniques.

Environmental noise, such as electromagnetic radiation or thermal fluctuations, can disturb the delicate quantum state of a qubit, leading to inaccuracies in measurement outcomes. Additionally, imperfections in hardware, such as imperfect qubit gates or faulty readout mechanisms, can introduce errors into the measurement process.

Moreover, the process of quantum measurement itself is not entirely deterministic, introducing intrinsic uncertainties into the measurement outcomes. The probabilistic nature of quantum mechanics means that even under ideal conditions, measurement outcomes can vary unpredictably.

Combating Errors in Quantum Computing

There are several techniques to mitigate errors. Some of them include:

(PDF) The first three-qubit and six-qubit full quantum multiple error-correcting codes with low quantum costs (researchgate.net)

Quantum Error Correction (QEC): This is the ultimate goal, allowing for complete correction of errors during computation. It works by encoding a single logical qubit (unit of information) across multiple physical qubits. This redundancy allows for the detection and rectification of errors that might occur in individual qubits.
Quantum Error Mitigation (QEM): This approach acknowledges the difficulty of full-fledged QEC in current noisy intermediate-scale quantum (NISQ) devices. Instead of actively correcting errors, QEM focuses on inferring the “noiseless” result from a noisy computation. Techniques like zero-noise extrapolation involve running the same circuit with slight variations and analyzing the results to extrapolate the ideal outcome with less noise.
Error Suppression: This proactive approach aims to minimize errors in the first place. By utilizing advancements in quantum control techniques, researchers can build hardware that is inherently more resistant to errors. This involves optimizing the design of qubits and gates to reduce their susceptibility to environmental noise and operational imperfections.
Fault-tolerant Architectures: These are theoretical designs for building large-scale quantum computers that can tolerate a certain level of errors without compromising the overall computation. These architectures often involve incorporating QEC codes and techniques to ensure fault tolerance, allowing for computations to proceed even with occasional errors.

In addition to these methods, recently Microsoft and Quantinuum have made significant strides in advancing the field of quantum computing. By integrating cutting-edge error correction techniques with state-of-the-art ion-trap hardware, the partnership achieved an 800-fold improvement in the logical error rate compared to physical qubits, as detailed in their recent publication on arXiv. This remarkable advancement was facilitated by Microsoft’s innovative qubit-virtualization system, seamlessly integrated with Quantinuum’s specialized hardware (Microsoft Quantum Blog) Notable points include:

800x Improvement: The logical error rate was reduced by 800 times compared to physical qubits, a substantial leap forward in error correction.
Active Syndrome Extraction: The successful demonstration of multiple rounds of active syndrome extraction represents a significant milestone in quantum error correction, enabling longer and more complex computations without failure.

Noisy Intermediate Scale Quantum (NISQ) Technology — Thomas J. Ackermann (bgp4.com)

This achievement could potentially pave the way for the quantum computing industry to enter the era of Intermediate-Scale Quantum (ISQ) computing, where reliable and scalable quantum systems can perform meaningful computations.