### Rise of hybrid quantum-classical computation

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## 2.3. Rise of hybrid quantum-classical computation

### Quantum measurements #

We have, until now, skipped over a crucial part of the quantum computation process: the role of quantum measurements. Quantum data, in isolation, is inherently inaccessible to us and the broader macroscopic world. A result from a quantum computation is only of value if we can probe it and get some readout value that we can display to the user or return to whoever launched the quantum computation.

Quantum physics measurements fundamentally differ from our classical understanding of just “reading out” data that is already there. This is the famous Schrödinger’s cat thought experiment of quantum mechanics: what data is within the qubits remains undefined until a measurement is performed. The act of observation will transform the quantum data: looking inside the box will, at random, either kill the cat or spare it1.

We thus need to add the measurement operation as a special case to our computer scientist’s model of quantum computing. Unlike purely quantum operations, measurements inherently involve interaction with the environment to produce a readout. Consequently, the no-delete and reversibility principles discussed earlier do not apply. Indeed, measurement is a lossy (and therefore irreversible) operation that projects the quantum state into one of a small subset of classical states. Which state the quantum state is projected into is non-deterministic. If one has access to an infinite supply of the same quantum state, then the whole state can be reconstructed by repeating measurements and analysing the distribution of outcomes2. Given no-cloning, however, this is unlikely to be the case, and so the full quantum result is hardly ever known. Instead, we must rely on well-designed measurement schemes to extract useful information from our partial access to the quantum states.

We model measurement as an operation that takes one qubit and outputs one purely classical bit3. In the circuit formalism, measurements are often implicitly added at the end of every qubit. Suppose we wish to make them explicit or add them elsewhere in the computation. In that case, we must introduce a graphical representation for the classical bit of data the measurement produces. The field has adopted the double-wire
for this, even though a “half” wire would arguably have been more appropriate to reflect the reduced information content relative to quantum wires. I present to you the measurement box:

### Measurements as first-class citizens #

It is very tempting to our feeble classical brains – and admittedly, we just did it ourselves in the previous paragraphs – to view measurements as merely a readout operation, an auxiliary operation that we are forced to perform at the end of a computation for operative reasons. This could not be further from the truth! In many ways, measurements are just as powerful tools as any other quantum operation – if not more so!

One eye-opening perspective on this is the field of measurement-based quantum computing (MBQC). Raussendorf and Briegel showed indeed Rausse., 2001Robert Raussendorf and Hans J. Briegel. 2001. A One-Way Quantum Computer. Physical Review Letters 86, 22 (May 2001, 5188--5191). doi: 10.1103/PhysRevLett.86.5188 that arbitrary quantum computations can be reproduced in the MBQC framework using only some resource quantum states that can be prepared ahead of time and measurements! In other words, given entangled qubits, measurements are all you need to perform quantum operations.

We will not explore MBQC further in this chapter (nor in this thesis, for that matter). Instead, we will use this as a motivation to explore what we can achieve with measurements. We have so far spared you from any mathematical alphabet soup. As we start discussing more concrete constructions of quantum computations, some introductory linear algebra and conventions around notation will become unavoidable.

> Dirac formalism. Quantum states are nearly unanimously written using kets: instead of referring to a quantum state as ψ\psi ψ, we write it wrapped in special brackets as ∣ψ⟩\ket{\psi}∣ψ⟩. This notation is also used when referring to the 0 00 and 1 11 states of qubits, written ∣0⟩\ket{0}∣0⟩ and ∣1⟩\ket{1}∣1⟩.

> Several states can be joined and considered together as one overall state. This is expressed using the tensor ⊗\otimes⊗ symbol: ∣ψ 1⟩⊗∣ψ 2⟩\ket{\psi_1} \otimes \ket{\psi_2}∣ψ 1​⟩⊗∣ψ 2​⟩ is the joint system of ∣ψ 1⟩\ket{\psi_1}∣ψ 1​⟩ and ∣ψ 2⟩\ket{\psi_2}∣ψ 2​⟩. When the states in question are all explicitly qubit states, we use the shorthand binary notation ∣0⟩⊗∣1⟩⊗∣0⟩=∣010⟩\ket{0} \otimes \ket{1} \otimes \ket{0} = \ket{010}∣0⟩⊗∣1⟩⊗∣0⟩=∣010⟩.

> We will introduce more notation along the way.

With this out of the way, let us look in more details at the first smart use of measurements: the block-encoding technique. Consider the following scenario: you would like to perform an operation A AA on an arbitrary quantum state ∣ψ⟩\ket{\psi}∣ψ⟩. Now, there are, unfortunately, many cases where implementing A AA as a quantum circuit made of primitive gates that can be executed on hardware is very expensive4.

However, what we can always do is express A AA as a matrix of dimensions 2 n×2 n 2^n \times 2^n2 n×2 n, where n nn is the number of qubits in the state ∣ψ⟩\ket{\psi}∣ψ⟩. Then, there is a neat trick that we can sometimes apply: instead of trying to execute A AA, we enlarge the matrix to a bigger A~\tilde{A}A~:

A~=(A G 1 G 2 G 3)\tilde{A} = \begin{pmatrix}A & G_1\\ G_2 & G_3 \end{pmatrix}A~=(A G 2​​G 1​G 3​​)

where G 1, G 2 G_1, G_2G 1​,G 2​ and G 3 G_3G 3​ are “garbage” matrices that we do not care about, but should combine into a matrix A~\tilde{A}A~ that we know how to execute on a quantum computer. Quantum computations must be matrices with a row and column number that is a power of two; so at a minimum, A~\tilde{A}A~ must be of size 2 n+1×2 n+1 2^{n+1} \times 2^{n+1}2 n+1×2 n+1, i.e. be a computation on n+1 n + 1n+1 qubits.

We restrict our considerations in the following to the case on m=n+1 m = n + 1m=n+1 qubits – other cases are similar. We thus need to add a qubit to our ∣ψ⟩\ket\psi∣ψ⟩ state to be able to pass it to our new operation. Such qubits that are added temporarily to facilitate a computation are a recurring feature in quantum computing and have thus earned themselves a name – ancilla qubits.

Let us take a look at the quantum states that result from executing A~\tilde{A}A~. If we add to ∣ψ⟩\ket \psi∣ψ⟩ a ∣0⟩\ket 0∣0⟩ ancilla state, our quantum operation acts as5

A~(∣0⟩⊗∣ψ⟩)=∣0⟩⊗A∣ψ⟩+∣1⟩⊗G 1∣ψ⟩.\tilde{A} (\ket 0 \otimes \ket \psi) = \ket 0 \otimes A \ket \psi + \ket 1 \otimes G_1 \ket \psi. A~(∣0⟩⊗∣ψ⟩)=∣0⟩⊗A∣ψ⟩+∣1⟩⊗G 1​∣ψ⟩.

The expression A∣ψ⟩A \ket \psiA∣ψ⟩ means the operation A AA applied to ∣ψ⟩\ket \psi∣ψ⟩ – exactly the output state we are seeking. If we input the ancilla qubit in state ∣1⟩\ket 1∣1⟩, we get garbage:

A~(∣1⟩⊗∣ψ⟩)=∣0⟩⊗G 2∣ψ⟩+∣1⟩⊗G 3∣ψ⟩.\tilde{A} (\ket 1 \otimes \ket \psi) = \ket 0 \otimes G_2 \ket \psi + \ket 1 \otimes G_3 \ket \psi. A~(∣1⟩⊗∣ψ⟩)=∣0⟩⊗G 2​∣ψ⟩+∣1⟩⊗G 3​∣ψ⟩.

So ∣0⟩⊗∣ψ⟩\ket 0 \otimes \ket \psi∣0⟩⊗∣ψ⟩ is definitely the input state we are more interested in.

How can we recover A AA from (1)? This is precisely what measurements do! When quantum states are expressed as sum of states, the terms of the sum form the possible measurement outcomes6. If we only measure a subset of the qubits, then the term corresponding to that measurement is isolated and all other terms disappear. Hence, if we measure the first qubit (that we introduced ourselves) in the zero state, then the remaining n nn qubits will be precisely in the desired state A∣ψ⟩.A \ket \psi.A∣ψ⟩. Success!

Using this “term isolating” property of measurements, known as state collapse, we can thus effect computations that would have been otherwise difficult or impossible to perform. There is however one important wrinkle that we cannot forget about: measurements are non-deterministic! We cannot assume that all measurements of the ancilla qubit will return the zero state. When ∣1⟩\ket 1∣1⟩ is measured on the ancilla, the remaining qubits are left in the G 1∣ψ⟩G_1 \ket\psiG 1​∣ψ⟩ state. The computation has thus failed, and the execution must be aborted and restarted. How often the block-encoding protocol that we have presented fails depends on the details of A AA and the choices of G 1, G 2 G_1, G_2G 1​,G 2​ and G 3 G_3G 3​ and is the main disadvantage of an otherwise very powerful quantum technique.

We will now explore two strategies to deal with “fails” in measurements. At the core of them is the idea of hybrid quantum-classical programs.

### Who said quantum computers could not fix their mistakes #

Failed computations are an expensive mistake in quantum computing as the no-cloning theorem prevents us from keeping a “backup” of the initial state. The fact that failures are in fact unlucky measurement outputs makes matters worse, given that measurements are the only irreversible quantum operation. It is therefore impossible in general to recover from a “wrong” measurement.

There are, however, prominent cases in which the computation can be corrected based on the measurement outcome, thus yielding deterministic results. Recall equation (1) of the previous section: there is a computation A AA on n nn qubits, that can be probabilistically computed using m=n+1 m = n + 1m=n+1 qubits using A~\tilde{A}A~:

A~(∣0⟩⊗∣ψ⟩)=∣0⟩⊗(A∣ψ⟩)+∣1⟩⊗(G∣ψ⟩),\tilde{A} (\ket 0 \otimes \ket \psi) = \ket 0 \otimes (A\ket\psi) + \ket 1 \otimes (G\ket\psi),A~(∣0⟩⊗∣ψ⟩)=∣0⟩⊗(A∣ψ⟩)+∣1⟩⊗(G∣ψ⟩),

for some “garbage” G GG. What if G GG is a reversible operation, i.e. there is an operation G−1 G^{-1}G−1 to undo G GG? Well then, we can still, at least in theory, recover A∣ψ⟩A\ket \psiA∣ψ⟩ by applying A∘G−1 A \circ G^{-1}A∘G−1:

G∣ψ⟩↦(A∘G−1)∘G∣ψ⟩=A∣ψ⟩,G \ket \psi \mapsto (A \circ G^{-1}) \circ G \ket \psi = A \ket \psi,G∣ψ⟩↦(A∘G−1)∘G∣ψ⟩=A∣ψ⟩,

but only if `1` was measured on the ancilla qubit7!

This is the beginning of quantum-classical hybrid computing: we start by performing quantum operations followed by measurements, the outcomes of which dictate what further quantum operations must be applied. We define for this purpose a classically controlled gate: a quantum operation that is only executed if a certain classical bit (the condition) is set. This bit will typically be a value derived from a previous measurement: it could be as simple as the outcome that a previous measurement yield, or a function of multiple past outcomes that must be evaluated on classical hardware (e.g. a CPU).

Mixing classical and quantum operations is a sure way to bring the quantum circuit representation to its knees. We adopt the following representation, in which a quantum gate that has an additional classical bit wire attached to it represents a classically controlled operation that is only executed if the bit value is `1`.

#### Quantum Teleportation #

Quantum teleportation is a simple example of performing classically controlled quantum operations to do circuit corrections based on measurement outcomes. It is also coincidentally one of the most fundamental protocols of quantum theory. Its name is slightly misleading. Think of it as data transfer for quantum data, with a mind-bending twist: at the time of the transfer, only classical data must be communicated between the sending and receiving parties. As a result of this protocol, quantum information can be transferred using plain old-school copper wires (or any other classical communication channels)!

This is predicated on one crucial action being performed before the start of the communication. For every qubit that should be transmitted, the parties must beforehand create and share among themselves a pair of qubits that will serve as the quantum resource during the protocol execution. This resource state is widespread enough that it got its name: the Bell pair state. It is written in Dirac notation as ∣00⟩+∣11⟩\ket{00} + \ket{11}∣00⟩+∣11⟩. As the notation indicates, it is a state with perfectly correlated measurements: when measured, the two qubits will always yield the same outcome, either both `0` or both `1`.

There turns out to be a straightforward circuit that maps the two-qubit ∣00⟩\ket {00}∣00⟩, which every two-qubit computation starts in, into the Bell pair state:

It is enough for us to think of it as a black box – or a grey box in this case.

We are interested in “teleporting” an arbitrary, single-qubit quantum state. Such a state can always be expressed as ∣ψ⟩=α∣0⟩+β∣1⟩\ket \psi = \alpha \ket 0 + \beta \ket 1∣ψ⟩=α∣0⟩+β∣1⟩, i.e. in the most general case, a one-qubit state will be in some superposition of the states ∣0⟩\ket 0∣0⟩ and ∣1⟩\ket 1∣1⟩. The paramenters α\alpha α and β\beta β are complex coefficients that encode the probabilities of measuring `0` or `1` – we can view them as the weights of a weighted sum.

We are now interested in combining a Bell resource state in a joint system with the arbitrary state ∣ψ⟩\ket \psi∣ψ⟩. The resulting three-qubit state is obtained with the ⊗\otimes⊗ operation, which distributes over sums just like usual multiplication:

(∣00⟩+∣11⟩)⏟first two qubits⊗(α∣0⟩+β∣1⟩)⏟third qubit= α∣000⟩+α∣110⟩+β∣001⟩+β∣111⟩.\begin{aligned} &\underbrace{(\ket {00} + \ket {11})}_{\text{first two qubits}} \otimes \underbrace{(\alpha \ket 0 + \beta \ket 1)}_{\text{third qubit}}\\=\ &\alpha \ket {000} + \alpha \ket {110} + \beta \ket {001} + \beta \ket {111}.\end{aligned}= ​first two qubits
(∣00⟩+∣11⟩)​​⊗third qubit
(α∣0⟩+β∣1⟩)​​α∣000⟩+α∣110⟩+β∣001⟩+β∣111⟩.​

We chose to place the Bell pair on the first two qubits and the arbitrary state on the third. The goal is to move the data that sits on that last qubit to the first qubit. Looking at the first qubit in the above expression, notice that the desired state ∣ψ⟩=α∣0⟩+β∣1⟩\ket \psi = \alpha \ket 0 + \beta \ket 1∣ψ⟩=α∣0⟩+β∣1⟩ appears in the first qubit if we can discard the second and third terms:

α∣0‾00⟩+(α∣110⟩+β∣001⟩)+β∣1‾11⟩\alpha \ket{\underline{\mathbf{0}}00} + {\color{gray}(\alpha \ket {110} + \beta \ket {001})} + \beta \ket{\underline{\mathbf{1}}11} α∣0​00⟩+(α∣110⟩+β∣001⟩)+β∣1​11⟩

This sounds very much like the measurement operations we have used before to isolate terms – but we need to isolate two terms simultaneously. We can resolve this issue by reorganising the expression8

α∣000⟩+α∣110⟩+β∣001⟩+β∣111⟩= (α∣0⟩+β∣1⟩)⏟=∣ψ⟩⊗(∣00⟩+∣11⟩)⏟Bell pair+(β∣0⟩+α∣1⟩)⊗(∣01⟩+∣10⟩)+(α∣0⟩−β∣1⟩)⊗(∣01⟩−∣10⟩)+(β∣0⟩−α∣1⟩)⊗(∣00⟩−∣11⟩)\begin{aligned}&\alpha \ket{000} + \alpha \ket{110} + \beta \ket {001} + \beta \ket {111}\\ =\ &\underbrace{(\alpha \ket 0 + \beta \ket 1)}_{= \ket \psi}\otimes \underbrace{( \ket {00} + \ket {11})}_{\text{Bell pair}}\\&+(\beta \ket 0 + \alpha \ket 1) \otimes (\ket {01} + \ket {10})\\&+(\alpha\ket 0 - \beta \ket 1) \otimes (\ket{01} - \ket {10})\\&+(\beta \ket 0 - \alpha \ket 1) \otimes (\ket {00} - \ket {11})\end{aligned}= ​α∣000⟩+α∣110⟩+β∣001⟩+β∣111⟩=∣ψ⟩
(α∣0⟩+β∣1⟩)​​⊗Bell pair
(∣00⟩+∣11⟩)​​+(β∣0⟩+α∣1⟩)⊗(∣01⟩+∣10⟩)+(α∣0⟩−β∣1⟩)⊗(∣01⟩−∣10⟩)+(β∣0⟩−α∣1⟩)⊗(∣00⟩−∣11⟩)​

Obtaining the ∣ψ⟩\ket \psi∣ψ⟩ state on the first qubit is thus as simple as isolating the first of these four terms. We do not know a priori how to measure ∣00⟩+∣11⟩\ket {00} + \ket{11}∣00⟩+∣11⟩ but we do know how to map that state to ∣00⟩\ket {00}∣00⟩: that’s the inverse of the Bell pair state preparation circuit! This results in the following circuit:

This brings us to the same situation as we had for the block encoding application above: conditioned on the measurement outcome of the second and third qubits being 0, the computation performs a state “teleportation”, moving ∣ψ⟩\ket \psi∣ψ⟩ from the third to the first qubit. We can compute the effect of Bell−1\textit{Bell}^{-1}Bell−1 on the overall expression of (2) to find all possible output states:

α∣000⟩+α∣110⟩+β∣001⟩+β∣111⟩↦Bell−1(α∣0⟩+β∣1⟩)⊗∣00⟩+(β∣0⟩+α∣1⟩)⊗∣01⟩+(α∣0⟩−β∣1⟩)⊗∣10⟩+(β∣0⟩−α∣1⟩)⊗∣11⟩\begin{aligned}\alpha \ket {000} + \alpha \ket {110} + \beta \ket {001} + \beta \ket {111} \overset{\textit{Bell}^{-1}}{\mapsto}&(\alpha \ket 0 + \beta \ket 1) \otimes \ket {00} \\&+ (\beta \ket 0 + \alpha \ket 1) \otimes \ket {01} \\& + (\alpha \ket 0 - \beta \ket 1) \otimes \ket {10} \\&+ (\beta \ket 0 - \alpha \ket 1) \otimes \ket {11}\end{aligned}α∣000⟩+α∣110⟩+β∣001⟩+β∣111⟩↦Bell−1​​(α∣0⟩+β∣1⟩)⊗∣00⟩+(β∣0⟩+α∣1⟩)⊗∣01⟩+(α∣0⟩−β∣1⟩)⊗∣10⟩+(β∣0⟩−α∣1⟩)⊗∣11⟩​

As expected, we do get ∣ψ⟩\ket \psi∣ψ⟩ on the first qubit for the measurement `00` (corresponding to the state ∣00⟩\ket {00}∣00⟩), but as it stands, this only has a 1 4\frac{1}{4}4 1​ probability of success.

You might notice, however, that the other states in which the first qubit can end up look remarkably similar, up to some sign flips and swaps ∣0⟩↔∣1⟩\ket 0 \leftrightarrow \ket 1∣0⟩↔∣1⟩. In particular, all states still have the amplitudes α\alpha α and β\beta β somewhere, so it does not seem unfathomable that these “wrong” states can be mapped back to ψ\psi ψ.

We can use the measurement outcomes of the second and third qubit to infer which of the “mistakes” occurred, and hence what state the first qubit has ended in. The `01` measurement outcome, for instance, results in the β∣0⟩+α∣1⟩\beta \ket 0 + \alpha \ket 1 β∣0⟩+α∣1⟩ state – this is just a bit flip away from ∣ψ⟩\ket \psi∣ψ⟩! This gate is known as X X X. Its colleague the Z Z Z gate on the other hand leaves ∣0⟩\ket 0∣0⟩ states untouched but flips the sign of ∣1⟩.\ket 1.∣1⟩. This would fix the `10` outcome. Finally, `11` requires both a `Z` and a `X` correction.

Putting these observations together, we can leverage classically controlled operations to obtain a fully deterministic protocol! The correct circuit implementing quantum teleportation is given by

In the scenario where a first party (Alice) wants to send a one-qubit quantum state to Bob, they can achieve that by creating a Bell pair state, the first qubit of which is given to Bob and the second to Alice. When Alice then gets in possession of another qubit ∣ψ⟩\ket \psi∣ψ⟩ whose data she wants to transmit to Bob, she can achieve that by executing Bell−1\textit{Bell}^{-1}Bell−1, measuring her two qubits and communicating the (classical) measurement outcomes to Bob. Bob can perform the necessary corrections and will then have state ∣ψ⟩\ket \psi∣ψ⟩.

It is beautiful and often overlooked how one of the most fundamental protocols of quantum information theory is, in fact, a hybrid classical-quantum computation. Quantum teleportation without classical communication is physically impossible: it would let Alice communicate with Bob instantly, even though he could be light years away – in other words, it would fundamentally break relativity.

### Repeat until success: If you fail, retry! #

Classical computer science has a straightforward solution whenever probabilistic computations that can fail are used: probability amplification or boosting Scheid., 2018Christian Scheideler. 2018. Probability amplification. Retrieved lecture notes, online, visited 30/12/2024 from https://cs.uni-paderborn.de/fileadmin-eim/informatik/fg/ti/Lehre/SS_2018/AA/lecture_5.pdf. The idea is so simple that it barely deserves a name: execute several independent runs of the computation and choose the most common outcome. If the probability of failure is below a certain threshold (e.g. 50% for a binary output), then with basic statistics, one can extrapolate the number of runs required to obtain any desired accuracy9.

We have been ignoring this approach so far since no-cloning prohibits us from repeating a procedure more than once on an input state ∣ψ⟩\ket\psi∣ψ⟩. However, in the scenario that the computation should only be executed on a specific, known input state and the computation that prepares that state is known, we can recover from computation failures by just preparing a new state identically.

Suppose we know how to execute the quantum computation P P P mapping

∣0⋯0⟩↦P∣0⋯0⟩=∣ψ⟩.\ket{0\cdots 0} \mapsto P \ket {0\cdots0} = \ket \psi.∣0⋯0⟩↦P∣0⋯0⟩=∣ψ⟩.

As before, we would like to compute A AA given an implementation of the computation A~\tilde{A}A~ that acts on a n nn-qubit state ∣ψ⟩\ket\psi∣ψ⟩ and an ancilla qubit in the ∣0⟩\ket 0∣0⟩ state. If the measurement of A~(∣0⟩⊗∣ψ⟩)\tilde{A}(\ket 0 \otimes \ket \psi)A~(∣0⟩⊗∣ψ⟩) returns `1`, then the computation failed. We can then discard all qubits and restart from the ∣0⟩\ket 0∣0⟩ state, applying P P P followed by A~\tilde{A}A~ and an ancilla measurement, repeating until we measure 0. As a pseudo-quantum circuit, we could express this as:

    psi_qs = create_qubits(n)
    while True:
       ancilla_q = create_qubit()
       obtain measurement m from:

        if m == 0:
            break  # success! we can exit loop and proceed
        else:
            reset_qubits(psi_qs)

At each iteration, we can either exit the loop if the state collapse was successful (`m == 0`), or reset the qubits to zero and try again. But pseudo circuits do not run on hardware! The only way to express this computation as an actual circuit is to unroll the loop, i.e. repeat the block within the loop as many times as we expect might be necessary10. The first two iterations would look as follows:

It should be obvious why we haven’t unrolled the loop any further – it quickly becomes unweildy. The resulting program is not only hard to display and read, but it also suffers from fundamental issues in practice. For one, the program size becomes hugely bloated, and beyond slowing down the compiler, it will also cause a host of issues on the control hardware in real-time, such as long load times, inefficient execution, and low cache efficiency.

Even more worryingly, when picking the maximum number of iterations, we face an impossible tradeoff: if the number of iterations is small, then the probability of failure will remain non-negligible. As we scale this value up, however, we are introducing more and more gates into the program to cover the odd case of multiple successive repeated failures. We do not intend to execute these gates on most computation runs. They come at a significant cost to the runtime. For each gate listed in the circuit, the condition for the gate’s execution must be checked, whether or not the gate ends up being executed. Furthermore, hardware schedulers might be forced to be pessimistic and schedule a time window for all conditional operations ahead of time. This will significantly delay any operation to be performed after the loop.

We, therefore, argue that the quantum circuit model is ill-suited as the representation for quantum programs that combine classical and quantum data. Such programs, however, are a fundamental building block towards developing meaningful large-scale quantum computations and are bound to become the norm. Beyond the examples discussed above—–including block-encodings, repeat-until-success schemes, distributed quantum computing and measurement-based quantum computing – one application of hybrid quantum-classical operations stands out as critically important for the large-scale deployment of quantum computing: quantum error correction (QEC) schemes. We discuss this use case in the next section.

1. It is ironic that Schrödinger’s thought experiment Schrö., 1935Erwin Schrödinger. 1935. Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaftern, intended to highlight the absurdity of quantum mechanics, has become the field’s most famous PR campaign. Sorry to disappoint – you won’t find felines occupying multiple states of existence (though qubits do!) ↩︎

2. This is known as state tomography Allahv., 2004A. E. Allahverdyan, R. Balian and Th. M. Nieuwenhuizen. 2004. Determining a Quantum State by Means of a Single Apparatus. Physical Review Letters 92, 12 (March 2004, 120402). doi: 10.1103/PhysRevLett.92.120402. One must perform measurements in multiple bases, i.e., different choices of classical states to project to. ↩︎

3. Where did the qubit go? All the information in a qubit post-measurement is also contained in the classical bit of output data – it is, therefore, redundant and renders the qubit useless. In our model, we, therefore, bundle measurement and qubit discard into one operation. ↩︎

4. or outright impossible, in cases where A AA is not a unitary linear operation, for example. ↩︎

5. This is obtained by a simple matrix multiplication. The vector representation of the quantum state ∣0⟩⊗ψ\ket 0 \otimes \psi∣0⟩⊗ψ is obtained using the Kronecker product. You can also just trust me that this works out this way. ↩︎

6. This is simplifying slightly. There is a necessary condition for this to be a valid measurement: the states in the sum must form a measurement basis, i.e. they must be orthogonal. This is satisfied here. ↩︎

7. Notice that, informally, we would hope to get a computation G GG such that G≈A G \approx AG≈A in the sense that it should somehow be closely related to A AA. This way, the resulting correction A∘G−1≈I d A \circ G^{-1} \approx IdA∘G−1≈I d would be close to the identity, and would be cheap to compute. ↩︎

8. Apologies, it seems at this point that we are conjuring up a complex expression out of nowhere. It is in fact just a change of basis – plain old linear algebra. The formula can be obtained easily by writing out the basis change matrix. ↩︎

9. This is fiendishly effective: the Hoeffding bounds guarantee that the probability of success will converge to 1 exponentially with the number of runs. ↩︎

10. In other words, we must pick a constant M M M for the maximum number of times we expect the loop to be executed. If a single loop iteration has a failure probability of p pp, the failure probability of the program with M M M unrolled iteration is then p M p^Mp M. ↩︎

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