Quantum Memories

A quantum memory accepts a state, preserves its quantum information for a useful interval, and returns it on a specified carrier. The ideal write–store–read process is the identity channel, possibly followed by a known change of basis, frequency, or encoding. Long coherence in an isolated spin is insufficient if writing and reading destroy the state. Excellent fidelity among rare retrieved events is also different from efficient storage.

This page develops the hardware mechanisms and the measurements that distinguish efficiency, retained coherence, noise, bandwidth, mode capacity, and controllable release time. Quantum Repeaters uses these memory properties to synchronize links; Quantum Network Architectures combines the devices into networks.

Helpful background. Bits, Qubits, Qudits, and Modes distinguishes encodings. Quantum Channels and Noise explains completely positive maps and their composition. Quantum Teleportation shows how entanglement and a classical record transfer an unknown state.

The write–store–read channel

Write the end-to-end memory map at storage time $t$ as

$$\mathcal M_t = \mathcal R \circ \mathcal S_t \circ \mathcal W.$$

Here $\mathcal W$ writes the input carrier into the storage degree of freedom, $\mathcal S_t$ describes storage plus applied control, and $\mathcal R$ returns an output carrier. The ideal target may include a known unitary $U_t$, frequency conversion, or a change from a photonic to a stationary encoding and back.

A simple flagged-erasure model is

$$\mathcal M_t(\rho) = \eta(t) U_t\rho U_t^\dagger + \left[ 1-\eta(t) \right] \lvert e\rangle\langle e\rvert,$$

where $\lvert e\rangle$ is orthogonal to the computational output space. This model cleanly separates efficiency $\eta$ from the state quality on successful retrieval. Real memories add dephasing, state-dependent loss, background photons, leakage, mode distortion, and imperfect erasure flags. For a one-photon dual-rail encoding, vacuum is outside the code and can represent an erasure. For a single-rail encoding that uses vacuum as logical zero, photon loss is not automatically an orthogonal erasure; see Erasure and Loss Channels.

A memory is more than a delay

A passive delay line implements a predetermined release time. A fully on-demand memory permits a controller to choose the read time after the write event, within a stated acceptance window. Intermediate devices include:

• fixed-delay fibre, cavity, or echo buffers;
• switchable loops with release only at discrete circulation times;
• memories with a fixed rephasing delay plus a programmable spin-wave dwell;
• random-access banks with independently addressable cells;
• stationary qubits that remain in place while the processor schedules later gates.

Each can be useful. The name should state which timing service is actually available.

Efficiency, fidelity, and useful storage time

Compare devices at the same input/output modes and storage time. Efficiency describes how often the intended output is available; fidelity describes the output state relative to a specified target. Bandwidth and mode capacity describe what can be accepted, while preparation, readout, and reset determine how often the device can be reused. These quantities are coupled: longer storage, stronger control, and more modes can change the noise as well as the retrieval probability.

Efficiency

For an optical memory, a common end-to-end definition is

$$\eta(t) = \frac{N_{\rm retrieved}(t)} {N_{\rm accepted\ input}},$$

with signal counts referred to declared spatial, spectral, temporal, and polarization modes. Noise subtraction can estimate this signal efficiency, but the raw output, including background, determines the actual retrieved state. An intensity ratio for coherent pulses is not automatically a success probability for an arbitrary qubit. For a single-photon retrieval process, one can resolve the conditional stages:

$$\eta = \eta_{\rm in} \eta_{\rm write} \eta_{\rm store}(t) \eta_{\rm read} \eta_{\rm out},$$

where the product notation means that each factor is a conditional efficiency. Correcting away coupling, filtering, or detector loss can help diagnose a component but does not describe the state available to the next network or processor module.

Conditional fidelity weights the accepted ensemble

Let $\mathcal E_{\rm s}$ be the completely positive, trace-nonincreasing map for a declared success event, including the correction of a known target unitary. For input $\psi=|\psi\rangle\langle\psi|$, define

$$\begin{aligned} p_{\rm s}(\psi)&=\operatorname{Tr}\mathcal E_{\rm s}(\psi),\\ \rho_{\rm s}(\psi)&=\frac{\mathcal E_{\rm s}(\psi)}{p_{\rm s}(\psi)}. \end{aligned}$$

For prior measure $d\mu(\psi)$, the average success probability and fidelity of the accepted ensemble are

$$\begin{aligned} \bar p_{\rm s}&=\int d\mu(\psi)\,p_{\rm s}(\psi),\\ F_{\rm cond}&=\frac{\int d\mu(\psi)\, \langle\psi|\mathcal E_{\rm s}(\psi)|\psi\rangle} {\bar p_{\rm s}}. \end{aligned}$$

Equivalently, each conditional fidelity is weighted by its success probability. Averaging the normalized fidelities equally over input states is a different statistic when success depends on the input. The normalized conditional map is then generally nonlinear, even though the unnormalized success map is linear. An experiment that samples equally from several prepared states must specify which of these averages it reports.

For a trace-preserving qubit channel $\mathcal M$ with identity target, define

$$\begin{aligned} |\Phi\rangle&=(|00\rangle+|11\rangle)/\sqrt2,\\ F_{\rm e}&=\langle\Phi| (\mathcal I\otimes\mathcal M)(|\Phi\rangle\langle\Phi|) |\Phi\rangle. \end{aligned}$$

For a uniform pure-qubit input ensemble, $F_{\rm avg}=(2F_{\rm e}+1)/3$ (Nielsen). Here the input and output are both the specified qubit space. For an erasure output in a larger space, one must retain the erasure branch or explicitly restrict the score to the qubit success map.

If $F_{{\rm e},{\rm s}}$ instead denotes overlap with the unnormalized Choi state of $\mathcal E_{\rm s}$, the uniform-input success-weighted overlap satisfies

$$\bar p_{\rm s}F_{\rm cond} =\frac{2F_{{\rm e},{\rm s}}+\bar p_{\rm s}}3.$$

The extra $\bar p_{\rm s}$ replaces the trace-preserving constant $1$. Thus using the deterministic formula on an unnormalized success branch overestimates the score. State-dependent acceptance need not invalidate this linear-map identity; it invalidates treating the normalized branch as a channel without qualification.

Classical benchmarks depend on the supplied states

For a Haar-uniform unknown pure qubit supplied as one copy, every measure-and-prepare memory has average fidelity at most $2/3$ (Massar–Popescu). Outcome-selective rejection does not improve this accepted-ensemble bound: each retained measurement outcome separately satisfies it, as Exercise 10 shows. The success-weighted classical bound is consequently $2\bar p_{\rm s}/3$.

The benchmark can change for a nonuniform prior, a finite test alphabet, or multiple copies. Weak coherent polarization pulses are particularly important: their multiphoton components provide several copies of the polarization state. A classical strategy can favor those components when loss is permitted. Calling such a pulse “single-photon level” does not make it a single-photon Fock state. Coherent-state and squeezed-state storage also require their own state ensembles and scores; Hammerer and colleagues develop the coherent-state benchmark.

A high conditional fidelity with a small retrieval probability can still preserve entanglement on the accepted branch. It does not imply a nearly deterministic identity channel. Conversely, scoring flagged erasures as zero overlap can put a useful quantum memory below $2/3$; the comparison must include the same acceptance convention on the classical side.

Storage time and lifetime

$T_1$, Ramsey $T_2^*$, echo $T_2$, and dynamically decoupled coherence time describe different experiments. The useful memory lifetime is the interval over which the complete write–read channel meets a task threshold.

For a qubit undergoing pure dephasing with coherence factor $\lambda(t)=e^{-t/T_2}$,

$$F_{\rm avg}(t) = \frac{2+\lambda(t)}{3}.$$

For simultaneous zero-temperature qubit amplitude relaxation with population survival $s(t)$ and independent additional dephasing factor $\lambda_\phi(t)$, the Bloch contractions are $\sqrt{s}\lambda_\phi$, $\sqrt{s}\lambda_\phi$, and $s$. The uniform-input fidelity is therefore

$$F_{\rm avg}(t)=\frac12+ \frac{2\sqrt{s(t)}\lambda_\phi(t)+s(t)}6.$$

Here $\lambda_\phi$ excludes the coherence decay already caused by relaxation; using a measured total $T_2$ as this extra dephasing would count relaxation twice. Control-pulse errors, state-dependent retrieval, and background counts require further terms.

Bandwidth and time–bandwidth product

The accepted bandwidth $B$ and storage time define

$$\mathcal B = B T_{\rm s}.$$

This time–bandwidth product indicates a scale separation, not a general bound or count of stored qubits. Temporal capacity depends on the allowed input window, pulse shape, guard intervals, control bandwidth, cross-talk, retrieval ordering, and signal-to-noise ratio. A long-lived single stationary qubit can have a large product without storing many simultaneous temporal inputs. Nunn and colleagues analyze protocol-dependent multimode capacity.

Noise and the single-photon regime

For a memory with unconditional output-noise probability $p_{\rm n}$ per accepted mode, a useful scale is

$$\mu_1 = \frac{p_{\rm n}}{\eta},$$

the mean input photon number that would give signal-to-noise ratio near one in a simple linear model. Noise must be measured with the same preparation, control, filtering, retrieval window, and detector gates used for storage.

Collective Optical Memories

Atomic ensembles enhance light–matter coupling by storing one optical excitation as a collective spin wave. A schematic single-excitation state is

$$\lvert1_{\rm s}\rangle = \frac{1}{\sqrt N} \sum_{j=1}^{N} e^{i\Delta\mathbf k\cdot\mathbf r_j} \lvert g_1\cdots s_j\cdots g_N \rangle.$$

The phase pattern $\Delta\mathbf k$ records the input and control wavevectors. Directional retrieval follows from collective interference. Atomic motion, field gradients, collisions, and spatially varying light shifts dephase this pattern.

For ballistic atoms with independent Gaussian velocity components of standard deviation $\sigma_v$ along $\Delta\mathbf k$, the coherent retrieval amplitude contains

$$\left\langle e^{i\Delta\mathbf k\cdot\mathbf v t}\right\rangle =e^{-(|\Delta\mathbf k|\sigma_v t)^2/2}.$$

The corresponding coherent retrieval efficiency decays as $e^{-(|\Delta\mathbf k|\sigma_v t)^2}$. The factor of two between amplitude and intensity exponents matters. This simple large-ensemble model neglects collisions, confinement, and control errors; nearly copropagating write fields reduce the spin-wave wavevector and this motional dephasing.

Electromagnetically induced transparency

In a three-level lambda system, a control field opens a transparency window for the signal and converts the propagating field into a dark-state polariton. Schematically,

$$\Psi_{\rm D} = \cos\theta\,\mathcal E - \sin\theta\,S,$$

where $\mathcal E$ is the optical mode and $S$ a collective spin coherence. Adiabatically reducing the control field rotates the polariton into matter; restoring the control retrieves it.

EIT memories can provide on-demand readout and excellent state fidelity. Their efficiency depends on optical depth, control shaping, decoherence, and mode matching. The transparency bandwidth can be narrow, and strong control fields can add leakage, fluorescence, and four-wave-mixing noise.

Off-resonant Raman storage

Raman memories detune the optical fields from the excited state and transfer the signal to a spin wave through a two-photon process. Large detuning can support broader bandwidth and reduce resonant absorption, but stronger control energy may be required. Spontaneous Raman scattering and four-wave mixing can dominate single-photon-level noise.

EIT and Raman operation are limiting regimes of a broader light–matter interface. The meaningful choice depends on bandwidth, optical depth, detuning, control power, and noise, rather than protocol name alone.

Gradient echo and controlled broadening

Controlled reversible inhomogeneous broadening maps different frequency components to different atomic detunings. Reversing the detuning gradient reverses dephasing and produces an echo. Gradient echo memory can offer high efficiency and pulse sequencing, especially in warm vapours and inhomogeneously broadened solids.

The reversal operation must preserve phase and avoid population inversion that would amplify spontaneous noise. A classical echo is not automatically a quantum memory; the retrieved field must beat the relevant classical benchmark and preserve nonclassical correlations.

Atomic-frequency-comb storage

An atomic-frequency-comb memory prepares a periodic absorption profile with tooth spacing $\Delta$ in hertz. Absorbed frequency classes dephase and then rephase at

$$t_{\rm echo} = \frac{1}{\Delta}.$$

The basic AFC echo is a fixed-delay memory. Applying a control pulse before the echo transfers optical coherence to a long-lived spin state; a later control pulse restores it and provides controllable spin-wave storage. If optical dephasing is suspended for a spin dwell $T_{\rm spin}$, ideal retrieval is delayed by $1/\Delta+T_{\rm spin}$ from the input, not simply by the time of the second control pulse. Each temporal input mode receives this delay; the pulse train is not compressed into a single echo time.

AFC memories are naturally temporally multimode because many pulses can be absorbed before the first pulse rephases. Rare-earth-doped crystals also offer broad inhomogeneous lines for spectral multiplexing. Spin-wave control adds noise and transfer loss, so fixed-delay mode-count records and on-demand single-photon records should not be compared as the same service.

Single Emitters and Small Registers

A single atom, ion, molecule, quantum dot, or colour centre can store a qubit in long-lived internal states. A cavity or nanophotonic resonator enhances the interaction with a selected optical mode. The cooperativity

$$C = \frac{4g^2}{\kappa\gamma}$$

uses an angular-frequency coupling $g$, cavity photon-number decay rate $\kappa$, and emitter excited-population decay rate $\gamma$. Pure dephasing is an additional limitation. Large cooperativity can improve reflection, absorption, and spin–photon gate probabilities, but useful performance also needs spectral stability, mode matching, collection, control, and reset.

Single-emitter memories offer:

• individually addressable qubits;
• local gates and nondestructive readout;
• heralded remote entanglement;
• compact registers with communication and storage qubits;
• direct integration with error detection.

Their bottlenecks include photon collection, indistinguishability, narrow bandwidth, spectral diffusion, fabrication variability, and probabilistic remote links.

Communication and storage qubits

A network node often separates a fast optical communication qubit from a long-lived storage qubit. An electron spin can couple strongly to a photon, while a nearby nuclear spin stores entanglement during later attempts. The local transfer and repeated optical excitation must not destroy the stored state.

Defect and Solid-State Spin Qubits explains optical transitions, nuclear registers, and their local transfer errors. Trapped-Ion Qubits and Neutral-Atom and Rydberg Qubits describe alternative stationary registers. For every family, coherence measured without optical attempts can differ from coherence during repeated communication operations.

Solid-State Ensemble Memories

Rare-earth ions in crystals combine narrow homogeneous transitions, large inhomogeneous bandwidth, long hyperfine coherence, and compatibility with photon-echo protocols. The crystal is stationary and naturally supports many spectral and temporal modes. Waveguides, cavities, and fibre doping can improve integration.

The costs are substantial:

• cryogenic operation;
• spectral-hole-burning preparation;
• narrow or material-specific wavelengths;
• magnetic-field stabilization;
• imperfect control-pulse transfer;
• fluorescence and free-induction-decay noise;
• spectral diffusion and instantaneous spectral diffusion;
• frequency conversion for telecom networks.

A long spin-coherence measurement, a bright-pulse optical echo, and storage of an unknown single-photon qubit test different properties. In particular, classical pulse interference can establish phase preservation in a tested mode without determining the noise and fidelity for single-photon inputs. The solid-state echo mechanisms are reviewed by Tittel and colleagues.

Microwave, Oscillator, and Acoustic Memories

Superconducting processors often use high-quality microwave cavities as storage modes and nonlinear qubits as controllers. A beam-splitter-like interaction between modes,

$$H_{\rm swap} = \hbar g \left( a^\dagger b+ab^\dagger \right),$$

with real $g>0$ transfers a state after an ideal interaction time

$$t_{\rm swap} = \frac{\pi}{2g}.$$

The transfer also produces a known phase. Starting with mode $b$ in vacuum,

$$|n\rangle_a|0\rangle_b \longmapsto(-i)^n|0\rangle_a|n\rangle_b.$$

An arbitrary superposition therefore needs the number rotation $e^{i\pi n_b/2}$ to recover its original coefficients. This is not a global phase except for a fixed number state. Detuning, unequal damping, finite pump bandwidth, and thermal occupation modify the ideal transfer.

High-quality cavities can separate the long-lived storage mode from its nonlinear controller (Reagor and colleagues). Bosonic Qubits explains oscillator encodings and repeated recovery; Ofek and colleagues gives an experimental example of active memory protection.

Other proposals and experiments use:

• spin ensembles coupled to microwave resonators;
• acoustic and mechanical modes;
• magnonic modes;
• long superconducting transmission lines;
• microwave-to-optical transducers connected to optical memories.

Strong coupling, state-transfer fidelity, thermal occupation, mode crowding, and transduction loss determine whether a long-lived mode is useful. Interconnects and Transduction explains conversion, mode matching, and added noise between different carriers.

Active and Error-Corrected Memories

A passive memory suppresses noise through isolation, clock transitions, material quality, or a Hamiltonian gap. An active memory repeatedly detects errors and applies recovery or updates a frame.

For a logical memory, the central benchmark is not physical $T_1$ or $T_2$. It is logical survival under repeated correction:

$$\epsilon_L(d,t) = 1-F_L(d,t),$$

measured versus code distance or another resource parameter $d$ at matched storage time and operation conditions. Decreasing error as resources grow supports fault suppression under the tested noise and operations. A finite-size trend alone does not prove an asymptotic threshold or fault-tolerant computation. Syndrome extraction, decoding, leakage, reset, and acceptance must be included in the comparison.

Dynamical decoupling is not quantum error correction. It averages selected Hamiltonian noise using open-loop pulses; it does not generally identify which error occurred. Conversely, an error-corrected memory can be harmed by noisy syndrome cycles even when its physical storage mode is excellent.

Dynamical Decoupling derives noise filtering; Dynamical Decoupling in Quantum Information studies its placement during idle windows. Terhal reviews error-corrected memories, while Metrics for Quantum Hardware explains comparison at matched storage times and workloads.

Photonic Loops and Buffers

Switchable fibre or integrated-waveguide loops can store a photonic qubit without converting it to matter. They can be broadband, encoding-agnostic, and simple to interface. If each circulation has transmission $\eta_{\rm c}$, then after $N$ cycles

$$\eta_N = \eta_{\rm in} \eta_{\rm c}^{N} \eta_{\rm out}.$$

Exponential loss limits long storage even if the surviving photons remain coherent. Polarization rotation, dispersion, switch leakage, phase noise, and timing jitter also accumulate. The scalar transmission model assumes the loop treats both rails of the chosen encoding equally; otherwise the surviving state can be filtered or distorted.

Multiplexing and Random Access

Memory multiplexing can use:

• temporal modes: many pulses before retrieval;
• spectral modes: independently addressed frequency bins;
• spatial modes: separate cells, wavevectors, or emitters;
• polarization modes: often implemented as two balanced rails;
• orbital or high-dimensional modes: qudits stored in a mode basis;
• hybrid modes: products of several degrees of freedom.

For $M$ independent attempts with success probability $p$, the probability that at least one succeeds is

$$p_{\geq1} = 1-(1-p)^M.$$

The benefit is real only if the successful mode can be identified, preserved, and routed with low cross-talk and acceptable latency. A total mode count must therefore include:

• per-mode efficiency and fidelity distributions;
• cross-talk matrix;
• simultaneous versus sequential addressability;
• fixed versus on-demand retrieval;
• mode-dependent storage time;
• demultiplexer loss;
• controller and detector throughput.

Waiting for heralds

The propagation time depends on geometry. If a photon crosses node separation $L$ and a classical message returns along the same fibre, the propagation contribution is $2L/v_g$. If both nodes emit toward a midpoint detector, the photons travel $L/2$ and its herald returns $L/2$ to each node, giving $L/v_g$. Local preparation, detection, and processing add latency in either case.

The useful lifetime must exceed the required wait, often through repeated attempts on another link. A rough comparison is $T_{\rm useful}/\tau_{\rm attempt}$, but the actual random waiting-time distribution is needed to average nonlinear fidelity decay. In general $\langle F(T)\rangle\ne F(\langle T\rangle)$.

Memories synchronize independent successes; they do not remove optical loss, imperfect Bell measurements, local gate errors, or waiting-time decoherence. Quantum Repeaters develops the scheduling and protocol calculations. Exercise 6 illustrates the ideal two-link limit.

Initialization, Write, Protection, and Readout

Initialization

Ensemble memories may require optical pumping, spectral hole burning, cavity locking, magnetic-field alignment, and a quiet period for fluorescence to decay. Single-emitter nodes require spin initialization, charge-state stabilization, frequency tuning, and nuclear-register preparation. Superconducting and mechanical memories require cooling and active reset.

Preparation time determines duty factor:

$$D = \frac{T_{\rm available}} {T_{\rm available}+T_{\rm prepare}}.$$

A high-efficiency memory with a long preparation dead time can have low service throughput.

Write and impedance matching

Efficient absorption requires temporal and spectral mode matching. In a linear reciprocal interface with the appropriate reversed control, optimal absorption can be related to time-reversed retrieval. This is an optimization statement under the model assumptions, not a guarantee that any exponentially shaped input is optimal. Cavities can suppress prompt reflection through impedance matching, while control-pulse shaping maps the optical envelope into the desired spin wave (Gorshkov and colleagues).

Quoted internal efficiency should not hide losses in mode preparation, frequency conversion, fibre coupling, or filtering.

Protection

Storage may use:

• magnetic shielding and clock transitions;
• spin echo or dynamical decoupling;
• optical or microwave dressing;
• decoherence-free subspaces;
• autonomous stabilization;
• repeated quantum error correction.

Every protection operation can add errors. Pulse area and phase errors, off-resonant excitation, heating, leakage, and cross-talk should be measured over the same duration used for the memory claim.

Read and reset

Readout may be destructive or nondestructive, heralded or unheralded, fixed order or random access. Retrieval should preserve the declared output mode and encoding. Reset must remove residual excitations and restore the prepared absorption or spin state without corrupting neighbouring cells.

Noise during storage and retrieval

Loss and erasure

Absorption failure, spontaneous emission into unwanted modes, cavity escape, conversion loss, filtering, and detector inefficiency reduce $\eta$. If loss is reliably flagged, it is an erasure; unflagged loss can be confused with vacuum inputs or detector failure.

Dephasing and inhomogeneity

Magnetic and electric noise, atomic motion, collisions, spectral diffusion, and coupling disorder randomize relative phase. Echoes reverse static or slowly varying inhomogeneity but do not reverse irreversible fluctuations.

Control-pulse noise

Strong optical control fields can leak through filters, produce fluorescence, drive four-wave mixing, or populate unwanted levels. Microwave and radio frequency pulses can heat the device, cause off-resonant rotations, and accumulate systematic error.

Mode distortion

Finite bandwidth and dispersion alter temporal envelopes. Unequal rail amplitudes can filter polarization or path qubits; unequal phases produce a coherent rotation. Spectral diffusion changes the retrieved frequency. A high state fidelity measured after optimizing an analysis mode may not equal overlap with the fixed mode expected by the next device.

Cross-talk

Multiplexed cells share control beams, cavities, pumps, and detectors. Writing or reading one mode can dephase another. A memory array should report a cross-talk matrix under simultaneous workloads, not only isolated-cell tests.

Background and false heralds

Detector dark counts, spontaneous Raman photons, pump leakage, fluorescence, and residual cavity population can produce an apparent retrieved event with no stored input. Background subtraction is useful diagnostically, but raw false-herald probability determines network behavior.

Nonstationarity

Optical depth, resonance frequency, control gain, detector efficiency, and magnetic field drift. Long data acquisitions can average over several memory channels. Interleaved calibration and time-resolved residuals are needed to show stationarity.

Technology Comparison

Platform	Write and read mechanism	Structural strengths	Central bottlenecks
warm atomic vapour	EIT, Raman, gradient echo, Faraday interaction	room temperature, broad beams, potentially high bandwidth	collisions, diffusion, control noise, four-wave mixing
cold atomic ensemble	EIT, Raman, collective emission	high optical depth, low noise, spatial multiplexing	vacuum and laser complexity, motion, cycle time
rare-earth-doped solid	AFC, CRIB, EIT, cavity enhancement	long spin coherence, temporal and spectral multiplexing	cryogenics, control transfer, narrow transitions, telecom conversion
single atom or ion	cavity absorption, Raman mapping, local gates	addressability, excellent state control, long coherence	collection, cavity coupling, narrow bandwidth, scaling
defect or quantum-dot register	spin–photon gate plus local storage transfer	integrated node, communication and nuclear memory qubits	spectral diffusion, fabrication spread, transfer fidelity
microwave or acoustic mode	resonant swap, parametric conversion	strong circuit coupling, bosonic encoding, processor integration	cryogenics, thermal noise, transduction, controller faults
photonic loop	switching and propagation	broadband, encoding-preserving, no matter interface	exponential circulation loss, discrete timing, phase drift
error-corrected logical memory	repeated syndrome extraction and recovery	scalable suppression in principle	ancilla, decoder, cycle error, overhead

There is no platform-independent winner. A broadband photonic synchronizer, an hour-scale transportable memory, a microsecond high-rate repeater node, and a processor logical memory solve different tasks.

Interpreting a memory comparison

A narrow-band long-lived register, a broadband photonic buffer, and a multimode ensemble solve different timing and encoding problems. Their best efficiency, lifetime, and bandwidth values cannot be multiplied into one hypothetical device unless those values can be achieved together.

Measure the input and output at the interfaces the next component uses. Report unconditional background and false heralds along with signal efficiency. A retrieved-mode fit optimized after the experiment can diagnose distortion, but the next device may require a fixed mode. Protection pulses and syndrome measurements must be running during a claimed protected-lifetime measurement.

The reviews by Lvovsky, Heshami, and their collaborators give complementary introductions to the mechanisms. Detailed comparisons of network experiments are in Network Case Studies.

Exercises

1. Separate efficiency from conditional fidelity

A memory retrieves $60\%$ of accepted qubits. The retrieved subset has conditional fidelity $0.99$, and loss is perfectly flagged. What is the success-weighted overlap $\eta F_{\rm cond}$? Why is it not a replacement for a full channel metric?

Solution

The success-weighted overlap is

$$\eta F_{\rm cond} = (0.60)(0.99) = 0.594.$$

The number combines two useful quantities but discards the operational value of the erasure flag and the task-dependent treatment of failed trials. A network may retry a flagged erasure, while a deterministic processor may not. The pair $(\eta,F_{\rm cond})$ plus the output state on failure is the more complete description.

2. Convert a dephasing time to average fidelity

For pure dephasing with $\lambda(t)=e^{-t/T_2}$, find the average qubit fidelity at $t=T_2$.

Solution

Substitution gives

$$\begin{aligned} F_{\rm avg}(T_2) &= \frac{2+e^{-1}}{3} \\ &\approx 0.789. \end{aligned}$$

A quoted $T_2$ is therefore not the time at which the memory remains nearly perfect. It is a decay constant whose acceptable fraction depends on the task.

3. Estimate a time–bandwidth product

A memory accepts $5\ {\rm MHz}$ bandwidth and stores for $100\ \mu{\rm s}$. Find $\mathcal B$. Does this prove it stores that many independent temporal qubits?

Solution

$$\mathcal B = (5\times10^6\ {\rm s^{-1}}) (100\times10^{-6}\ {\rm s}) = 500.$$

The result is a scale for possible temporal capacity. It does not include guard intervals, pulse-shape orthogonality, control bandwidth, noise, retrieval order, or cross-talk. Independent storage must be demonstrated.

4. Set an AFC echo time

An atomic frequency comb has tooth spacing $\Delta=200\ {\rm kHz}$. When is the first fixed echo expected?

Solution

With $\Delta$ expressed in cycles per second,

$$t_{\rm echo} = \frac{1}{\Delta} = \frac{1}{2.00\times10^5\ {\rm s^{-1}}} = 5.0\ \mu{\rm s}.$$

This release time is fixed by comb preparation. On-demand retrieval requires transfer to and from a spin wave or another controllable storage layer.

5. Compare two heralding geometries

Nodes are separated by $50$ km of fibre with group index $n_g=1.5$. Neglect local processing. Compare a photon sent to the far node with a returning herald, and two photons sent to a midpoint detector with heralds returned to both nodes.

Solution

The group velocity is $v_g=c/n_g=2.00\times10^8$ m/s. For the far-node detector,

$$\tau_{\rm far}=2L/v_g=0.50\ {\rm ms}.$$

For a midpoint detector, each node waits for $L/2$ of photon propagation followed by $L/2$ of classical propagation:

$$\tau_{\rm mid}=L/v_g=0.25\ {\rm ms}.$$

Both times are measured from emission. A memory written earlier waits longer; local delays and additional unsuccessful link attempts must also be included.

6. Quantify two-link synchronization

Each of two independent links succeeds with probability $p$ per attempt. Without storage, simultaneous success takes $1/p^2$ attempts on average. Let $X,Y$ be independent geometric waiting times on $1,2,\ldots$ for synchronous attempts of equal duration. Once a link succeeds, keep it while the other continues. With perfect memories, the expected number of clock rounds to obtain both is

$$\mathbb E[\max(X,Y)] = \frac{3-2p}{p(2-p)}.$$

Derive the formula from $\max(X,Y)=X+Y-\min(X,Y)$ and compare the two values for $p=0.01$.

Solution

Both means are $1/p$. A round contains at least one success with probability $1-(1-p)^2=p(2-p)$, so $\mathbb E[\min(X,Y)]=1/[p(2-p)]$. Subtracting gives the stated result. Without memory,

$$\frac{1}{p^2} = 10^4$$

attempts. With perfect storage,

$$\begin{aligned} \mathbb E[\max(X,Y)] &= \frac{3-0.02}{(0.01)(1.99)} \\ &\approx 150. \end{aligned}$$

Synchronization changes the small-$p$ scaling from order $p^{-2}$ to order $p^{-1}$. Real gains are smaller because the first success decoheres while waiting, and write, read, and swap operations are imperfect.

7. Evaluate multiplexed availability

One temporal mode succeeds with probability $p=10^{-3}$. What is the chance of at least one success across $M=100$ independent modes?

Solution

$$\begin{aligned} p_{\geq1} &= 1-(1-10^{-3})^{100} \\ &\approx 0.0952. \end{aligned}$$

The probability rises to about $9.5\%$. Routing loss, cross-talk, and a limited detector or controller rate can reduce the realized gain.

8. Compute the noise-equivalent input

A memory has unconditional noise probability $p_{\rm n}=2\times10^{-4}$ per output gate and end-to-end efficiency $\eta=0.20$. Find $\mu_1$.

Solution

$$\mu_1 = \frac{2\times10^{-4}}{0.20} = 1.0\times10^{-3}.$$

An input with mean photon number about $10^{-3}$ gives signal probability comparable to the background in the simple linear model.

9. Combine write, decay, and read

A single-photon memory writes with efficiency $0.80$, stores with survival $e^{-t/T}$, and reads with efficiency $0.85$. Find the total efficiency at $t=2T$.

Solution

$$\begin{aligned} \eta(2T) &= (0.80)e^{-2}(0.85) \\ &\approx 0.0920. \end{aligned}$$

Only about $9.2\%$ of accepted inputs emerge. Long conditional coherence does not compensate for low write/read efficiency or storage decay.

10. Can a classical memory improve the qubit benchmark by rejection?

The input is one copy of a Haar-uniform pure qubit. A measurement outcome has positive effect $E\ne0$, and the classical device prepares $|\phi\rangle$ after that outcome. Use

$$\int d\psi\,\psi\otimes\psi=\frac{I+S}{6},$$

where $S$ swaps two qubits, to show that this outcome’s conditional average fidelity cannot exceed $2/3$. Explain the implication for accepting only selected outcomes.

Solution

The outcome probability averaged over inputs is $\operatorname{Tr}E/2$. Its fidelity numerator is

$$\int d\psi\,\operatorname{Tr}(E\psi) \langle\phi|\psi|\phi\rangle =\frac{\operatorname{Tr}E+\langle\phi|E|\phi\rangle}{6}.$$

Their ratio is

$$F_E=\frac13\left(1+ \frac{\langle\phi|E|\phi\rangle}{\operatorname{Tr}E}\right) \leq\frac23,$$

because a positive matrix’s largest eigenvalue cannot exceed its trace. A rank-one effect and the corresponding output state attain the bound. Mixing retained outcomes or using mixed prepared states cannot improve it. Rejection can change the distribution of accepted states, but the correctly weighted accepted average still obeys $2/3$.

The single-copy premise matters. With $n$ identical copies of the unknown qubit, optimal estimation reaches $(n+1)/(n+2)$ (Massar–Popescu). A weak coherent pulse’s multiphoton sectors can therefore support a better classical score if the strategy accepts them preferentially.

11. Recover a superposition after a mode swap

Use the resonant swap Hamiltonian above with mode $b$ initially empty. What happens to $(|0\rangle+|1\rangle)_a/\sqrt2$ at $\pi/(2g)$? What correction restores the coefficients?

Solution

The one-excitation sector evolves as $|1,0\rangle\mapsto\cos(gt)|1,0\rangle-i\sin(gt)|0,1\rangle$, while vacuum is unchanged. The result is

$$|0\rangle_a\otimes\frac{|0\rangle_b-i|1\rangle_b}{\sqrt2}.$$

The number rotation $e^{i\pi n_b/2}$ multiplies $|1\rangle_b$ by $i$ and restores the input. It also corrects every coefficient of a general finite-energy superposition because the $n$-photon sector acquired $(-i)^n$.

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